Combinational Mathematics Part 1
|
|
- Delphia Dayna Ferguson
- 5 years ago
- Views:
Transcription
1 j1 Combinational Mathematics Part 1 Jon T. Butler Naval Postgraduate School, Monterey, CA, USA Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 1 Monterey Coast Pacific Grove Monterey Bay Acquarium Naval Postgraduate School Carmel Big Sur Monterey Points of Interest Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 4 Monterey Coast We are here I live here Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 2 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 5 San Francisco Pacific Grove Monterey Los Angeles Purple Ice Plant Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 3 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 6 1
2 Slide 1 j1 jbutler, 3/24/2010
3 Monterey Bay Acquarium Linus Pauling The only person to win two unshared Nobel Prizes chemistry and peace. He lived in Big Sur. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 7 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 10 Hotel Del Monte Naval Postgraduate School Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 8 Passwords Question? Which of these passwords are strong? tomoko2015 tomoko p@ssw0rd password airplane 5. $1ngle single 6. IgfMUiM2 Igraduate from Meiji University in March 2015 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 11 Clint Eastwood Carmel Carmel Mission Passwords Question? Which of these passwords are strong? % 2. tomoko % 3. p@ssw0rd 40% 76% 5. $1ngle 36% 6. IgfMUiM2 57% Score: RoboForm Password Meter ordmeter.com/ Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 9 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
4 Storing Passwords Computer Password File #6ag&kP0 Z +jq?xlb62 d= />LpV* :z<)6kxw3 Po7(!X&zJ K,6Q1/8kc Encryption easy to go from plaintext to ciphertext and easy to go from ciphertext to plaintext Harder to crack Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 13 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 16 Storing Passwords Storing Passwords Cracking Passwords It is too slow to try to break into a system by trying combinations at the accountpassword page. Hackers prefer to breakin once and steal the whole password file. Then, they can use a supercomputer to try many combinations. Computer Password File )?Hk2m3D =8Jc:fF9q #2JK/.Ksz p[=9hbz@ o_+v2wg L*wn;2?sa One-Way Hash easy to go from plaintext to cipher-text but hard from ciphertext to plaintext Very Hard to crack Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 14 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 17 Storing Passwords Computer Password File 1234 tomoko2015 $!ngle IgfMUiM2 Plaintext Easy to crack Assume a one-way hash In a brute-force attack, an attacker tries many passwords and sees if any of the encrypted forms matches one of the passwords he tried. For each match, he can access that person s account. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 15 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
5 Assume a one-way hash If all passwords are 4 digits, then the passwords are 0000 through There are only 10 4 = 10,000 passwords, and it is easy to try all of them. If the passwords are all 4 lowercase letters, then the passwords are between aaaa and zzzz. Now, there are 26 4 = 456,976 passwords. It is also easy to try all. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 19 Brute force times to crack all passwords A GPU can do. 20M hashes per second is 20,000,000 passwords per second. A CPU can do. a A 1 $ 38 days to crack all 8 character passwords! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 22 Example Hack LinkedIn, a social network website was hacked on June 5, 2012 by Russian cyberthieves. Nearly 6,500,000 million user accounts were hacked. The stolen passwords, were encryp-ted, but were decrypted andpostedonarussianpassword decryption forum later on that same day. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 20 Dictionaries People usually do not use random characters. Therefore, hackers often use dictionaries to test for patterns. The dictionary can even be the passwords previously hacked in a password file! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 23 Example Hack This provides an understanding of how people choose their passwords. 21% use only lowercase letters 44% use lowercase & numbers Thus, 65% choose a combination of lowercase letters and numbers. This is good news for hackers!! Dictionaries + Rules People often leetify dictionary words to produce passwords. Examples are 1. E or e 3 2. S or s $ 3. A or 4. I or i! 5. Add numbers to the end of words Hackers know these rules and can take dictionary words and apply the rules. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 21 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
6 How to choose a password 1.Choose random characters with uppercase letters, lower case letters, numbers, and special characters # $ etc.) 2.Choose long passwords password cracking time goes up exponentially with password length Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 25 Outline Combinatorial Math Part 1 Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 28 How to choose a password 3. Change password and use different passwords for different accounts 4. Use a password manager, like 1password ( or LastPass ( You need only remember one password. All of your passwords are stored on your computer or in the cloud. Yogi Berra Yogi Berra played baseball for the New York Yankees in Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 26 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 29 Outline Combinatorial Math Part 1 Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 27 Yogi Berra He is famous for Yogism s. 1. It is not over, until it is over. 2. Nobody goes there anymore. It is too crowded. 3. Baseball is 90% mental. The other half is physical. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
7 Yogisms for combinatorial mathematics 1. If it is impossible, there are 0 ways to do it. 2. If there is only one way to do it, then the number of ways to do it is There is one way to choose 0 objects from 0 objects. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 31 Example Suppose there are 5 red balls and 3 green balls. Then, there are 5+3 = 8 ways to choose one ball of any color Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part Outline Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 32 Outline Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 35 Rule of Sum Rule of Product If one event can occur in m ways and another event can occur in n ways, there are m+n ways exactly one event can occur. If one event can occur in m ways and another event can occur in n ways, there are mn ways the two events can occur together. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 33 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
8 Example Suppose there are 5 red balls and 3 green balls. Then, there are 15 ways to choose one red ball and one green ball Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 37 Example How many ways, P(n,n), are there to arrange n of n objects? By the rule of product, P(n,n) = n(n-1)(n-2) 2 1 = n!. There are n There are There are ways to n-1 ways to n-2 ways to choose the choose the choose the 1 st object. 2 nd object. 3 rd object. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 40 Example (cont d) Note: When we choose one red and one green ball, order is not important. For example, 1 2 = 2 1 Example This is n factorial. By the rule of product, P(n,r) =n(n-1)(n-2) (n r+1) = n! (n-r)! This is a permutation of r of n objects. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 38 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 41 Example Your PC has an address register with 24 bits. How many addresses are there? There are 24 bits, each with two values, 0 and 1. By the rule of product, there are = 2 24 = 16,777,216 addresses. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 39 P(n,r) = n(n-1)(n-2) (n-r+1) can be viewed as a distribution of n balls to r cells, as follows. There are n ways to fill the 1 st cell. There are n-1 ways to fill the 2 nd cell. There are n-2 ways to fill the 3 rd cell. There are n-r+1 ways to fill the r th cell. Note:Someballsmaybeleftover. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
9 Example How many permutations are there of two elements in S ={a,b,c,d}? There are12 ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, and dc. That is, n =4 and r =2in P(n,r) =n (n-1) = n!/(n-r)! = 12. Example Therefore, 2! times (the number of arrangements of ALL) = (number of arrangements of AL 1 L 2 ). That is, the number of arrangements of ALL is 3! = 3 2! These are ALL, LAL, and LLA. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 43 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 46 Remember: In a set S of n distinct objects, a permutation is an arrangement or ordering of objects from S. Permuting When There Are Objects of the Same Type If there are n 1 objects of the first type, n 2 objects of the second type,, n r objects of the r th -type, where n = n 1 + n 2 + +n r, then the number of arrangements of all n objects isiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii n! n 1! n 2! n r! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 44 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 47 Example How many ways are there to arrange the three letters in ALL? If all letters were distinct, like AL 1 L 2, there would be 3!=6, by the rule of product. But, two letters are the same! h ALL <=>AL 1 L 2 AL 2 L 1 LAL <=>L 1 AL 2 L 2 AL 1 LLA <=>L 1 L 2 A L 2 L 1 A Example In our previous example, n =3n 1 =1and n 2 = 2. Therefore, the number of arrangements of the letters of ALL is Thisis3. 3! 1! 2! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 45 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
10 Permuting When There Are Objects of the Same Type n! n 1! n 2! n r! is called a multinomial. n! C(n,r) = r! (n-r)! is a binomial and is the number of ways to choose r from n objects. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 49 Circle Permutations Choose one person, say A, to be at the top. Then, choose the order of the remaining people in 5! = 120 ways. Thus, the number of circular permutations on 6 objects is 5!. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 52 Outline Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 50 Circle Permutations Suppose that A, B, and C are girls, while D,E,andFareboys. Wewanttoarrange them so that boys and girls alternate. How many ways are their to do this? Choose A, a girl to be in the top position. There are 3! ways to arrange the boys and 2! ways to arrange the remaining girls or 3! 2! = 12 ways. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 53 Circle Permutations If six people, A, B, C, D, E, and F are seated around a round table, how many circle arrangements are possible? A E A A F B D F B F B = = E C C A C E C F D B D D (a) (b) (c) (d) (a) and (b) are identical, while (c) and (d) are different. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 51 E Recall there are 3 ways to arrange ALL We want to count the ways to choose 2 objects from 3. Let L be the chosen ojects and A the unchosen object. Thus, the three ways to choose are ALL, LAL, and LLA. The number of ways to choose 2from3objectsis iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii 3! = 3. 1!2! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
11 Recall there are 3 ways to arrange ALL From this, we can observe n! P(n,r) C(C(C(n,r) = ii = r!(n-r)! r! C(C( P(n,r) = ir! C(n,r) The set of permutations is just the set of combinations with all chosen objects permuted in some way. Pascal s Rule - Example Consider 4 objects C(4,2) = C(3,2) + C(3,1) = = 6 {b,c}, {b,d}, {c,d} or {a,b}, {a,c}, {a,d} a is the special object Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 55 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 58 Outline Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 56 Pascal s Triangle Sum Diff Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 59 Pascal s Rule By the Rule of Sum C(n,r) =C(n-1,r) +C(n-1,r-1) Choose r objects from n-1 objects not including a special object Choose the special or object and r-1 objects from n-1 objects not including the special object Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 57 Pascal s Triangle r n 0 Sum Diff Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
12 1. If it is impossible, thereare0waystodo Pascal s Triangle it. (choose 2 objects from 1 - impossible). 3. There is one way to choose 0 objects from 0 objects. r 2. If there is only one n 0 way to do Sum it, then Diff. the number of ways to do it is 1 (choose objects from2 1) Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 61 Outline Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 64 Pascal s Triangle Sum Diff. Pascal s Rule C(3,2) + C(3,1) = C(4,2) Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 62 Combinations With Repetition Question: In how many ways can you choose r objects from n objects when repetitions are allowed? That is, now one can choose an object 0, 1, 2, etc. times. Before, we could only choose 0 or 1 times. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 65 Pascal s Triangle This was first published in 1653 by Blaise Pascal (from France). These numbers were known more than a 1000 years earlier. Pascal s version of the triangle. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 63 Combinations With Repetition Example: The number of ways to choose r = 2 objects, with repetition, from n = 3 is 6. Consider {a,b,c}. One can choose 2 objects as follows {a,a}, {a,b}, {a,c}, {b,b}, {b,c}, and {c,c}. The number of ways to choose without repetition is 3, (the blue ways). Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
13 Combinations With Repetition Answer: A choice of r objects with repetition can be viewed as follows. Consider n + r 1 objects separated by r 1lines. The non-selected lines represent choices of r objects with repetition. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 67 Outline Introduction Rule of Sum Rule of Product Circle Permutations Pascal s Triangle Choice With Repetition Examples Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 70 Combinations With Repetition Answer: In our example, we had r =2andn = 3. So, among n+r-1 = 4 objects choose r =2 lines. {a,a} {a,b} {b,b} {a,c} {b,c} {c,c} Naming the Ace Paradox It appears to be wrong, but it is not! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 68 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 71 Combinations With Repetition Summary: The number of ways to choose r objects from n with repetition is C(n+r 1, r) =C(n+r-1,n-1) A Deck of Cards (52 cards) Each card has a 1) number and a 2) suit. 1) Number A KQJ Ace Jack Queen King 2) Suit Clubs Hearts Spades Diamonds Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 69 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
14 A Deck of Cards A K Q J Cards Clubs Spades Diamonds Naming the Ace Question: Why does naming the ace increase the probability of having another ace? Hearts Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 73 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 76 Definitions 1. To deal 52 cards to 4 people means to randomly give 13 cards to each of 4 people. Usually, the people know only those cards given to them. 2. A hand is a set of 13 cards that one deals to a person. Use a Smaller Set Give 2 cards to each of 2 people Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 74 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 77 Naming the Ace 1. Deal 52 cards to 4 people. One person says I have an ace. What is the probability she/he has another ace? Answer: 5359/14498 < 50%. 2. Deal 52 cards to 4 people. One person says I have an ace of spades What is the probability she/he has another ace? Answer: 11686/20825 > 50%. Below are the six different hands with two cards Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 75 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
15 Naming the Ace There are 6 ways a person can have a hand of 2 cards, as shown. If someone says, I have an ace, the probability she/he has another ace is 1/5 (there are 5 hands with at least one ace, only one of which has another ace). There are four equally likely outcomes Older Younger Boy Boy Boy Girl Girl Boy Girl Girl (impossible) Since only one of three possible outcomes consists of two boys, the probability is 1/3! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 79 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 82 Naming the Ace If he/she says, I have an ace of spades, then the probability she/he has another ace is 1/3. Naming the ace reduces the possibilities, and thus increases the probability! Playing Keno Keno is a game in which you choose 6 numbers from 1 to 80 and the casino chooses 20 numbers from 1 to 80. You enter your choice of numbers for 100 yen. The casino pays you back depending on how many matches occur. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 80 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 83 Children Paradox Mr. Yamada has two children. At least one is a boy. What is the probability that the other is a boy also? Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 81 Playing Keno For example, if all 6 of your numbers match numbers chosen by the casino, you win 100,000 yen! Question: What is the probability P(k) that k of the numbers you chose match numbers chosen by the casino? Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
16 Playing Keno Number of ways k numbers can match. P(k) = C(80,k)C(80-k,20-k)C(60,6-k) C(80,20)C(80,6) Number of ways casino can choose 20 numbers. Number of ways casino can choose the rest of its numbers. Number of ways person can choose the rest of its numbers. Number of ways person can choose 6 numbers. How to Become Rich by Playing Keno You cannot. Spend your money somewhere else. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 85 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 88 Playing Keno Exp. the results of an experiment done in a class of 40 students. Each student chose 6 numbers and the teacher choose 20. k P(k) Exp Yen Paradox A coin has two sides Head (H) and Tails (T). In this game, a coin is flipped producing H or T. To play, you pay 100 yen. If it is T, the coin is flipped again. If it is H, you receive some money, as shown on the next slide. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 86 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 89 Probability Playing Keno Probability of matching k numbers k Probability Experiment Exact Yen Paradox Outcome You receive H TH TTH TTTH TTTTH TTTTTH 100 yen 200 yen 400 yen 800 yen 1600 yen 3200 yen Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 87 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
17 Yen Paradox Of course, if you pay 100 yen, you should play because you will always receive at least 100 yen. However, should you play if it costs 10,000 yen instead of 100 yen? Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 91 Paths in a Chessboard Make 7 EAST moves and 7 NORTH moves. Once you make the EAST moves, the rest are NORTH moves. There are C(14,7) = 3,432 ways to choose the EAST moves. Thus, there are 3,432 different paths. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 94 Yen Paradox You can expect to receive 1_ 1_ You should pay ANY amount to play!! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 92 Birthday Paradox Answer: Compute the probability that no two people among n people will have the same birthday. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 95 Paths in a Chessboard You are here. Go here. Only go EAST or NORTH. How many paths are there? Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 93 Birthday Paradox That is, there are 365 n ways n people can have birthdays, but P(365,n) = (365-n+1) ways n people can all have different birthdays. Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
18 Birthday Paradox Thus, the probability we seek is (365-n+1) 365 n P(n) =1 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 97 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part Birthday Paradox n P(n) Note: At n =23, the probability is greater than 50%! Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part 1 98 Next Talk Combinatorial Mathematics Part 2 Friday, December 18 Meiji Univ. 10:30-12:00 October 9, 2015 J. T. Butler Combinatorial Mathematics Part
Combinational Mathematics - I
Combinational Mathematics - I Jon T. Butler Naval Postgraduate School, Monterey, CA, USA We are here I live here Meiji University 10:50-12:30 September 28, 2018 J. T. Butler Combinatorial Mathematics I
More informationDiscrete Mathematics: Logic. Discrete Mathematics: Lecture 15: Counting
Discrete Mathematics: Logic Discrete Mathematics: Lecture 15: Counting counting combinatorics: the study of the number of ways to put things together into various combinations basic counting principles
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationTopics to be covered
Basic Counting 1 Topics to be covered Sum rule, product rule, generalized product rule Permutations, combinations Binomial coefficients, combinatorial proof Inclusion-exclusion principle Pigeon Hole Principle
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
More information1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
More informationFinite Math - Fall 2016
Finite Math - Fall 206 Lecture Notes - /28/206 Section 7.4 - Permutations and Combinations There are often situations in which we have to multiply many consecutive numbers together, for example, in examples
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More informationSecret Key Systems (block encoding) Encrypting a small block of text (say 128 bits) General considerations for cipher design:
Secret Key Systems (block encoding) Encrypting a small block of text (say 128 bits) General considerations for cipher design: Secret Key Systems (block encoding) Encrypting a small block of text (say 128
More informationCombinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
More informationSection : Combinations and Permutations
Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More informationPermutations and Combinations
Motivating question Permutations and Combinations A) Rosen, Chapter 5.3 B) C) D) Permutations A permutation of a set of distinct objects is an ordered arrangement of these objects. : (1, 3, 2, 4) is a
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationJIGSAW ACTIVITY, TASK # Make sure your answer in written in the correct order. Highest powers of x should come first, down to the lowest powers.
JIGSAW ACTIVITY, TASK #1 Your job is to multiply and find all the terms in ( 1) Recall that this means ( + 1)( + 1)( + 1)( + 1) Start by multiplying: ( + 1)( + 1) x x x x. x. + 4 x x. Write your answer
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationAlgebra II- Chapter 12- Test Review
Sections: Counting Principle Permutations Combinations Probability Name Choose the letter of the term that best matches each statement or phrase. 1. An illustration used to show the total number of A.
More informationCounting (Enumerative Combinatorics) X. Zhang, Fordham Univ.
Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 Chance of winning?! What s the chances of winning New York Megamillion Jackpot!! just pick 5 numbers from 1 to 56, plus a mega ball number
More informationCS1800: Permutations & Combinations. Professor Kevin Gold
CS1800: Permutations & Combinations Professor Kevin Gold Permutations A permutation is a reordering of something. In the context of counting, we re interested in the number of ways to rearrange some items.
More informationMath 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
More informationFinite Mathematics MAT 141: Chapter 8 Notes
Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication
More informationGeneralized Permutations and The Multinomial Theorem
Generalized Permutations and The Multinomial Theorem 1 / 19 Overview The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 2 / 19 Outline The Binomial Theorem
More informationCISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest
More informationElementary Combinatorics
184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are
More informationChapter 4: Introduction to Probability
MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 1 6
Math 300 Exam 4 Review (Chapter 11) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Give the probability that the spinner shown would land on
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 22 Fall 2017 Homework 2 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 1.2, Exercises 5, 7, 13, 16. Section 1.3, Exercises,
More informationCS1800: More Counting. Professor Kevin Gold
CS1800: More Counting Professor Kevin Gold Today Dealing with illegal values Avoiding overcounting Balls-in-bins, or, allocating resources Review problems Dealing with Illegal Values Password systems often
More informationNovember 11, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.
More informationMathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015
1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:
More informationProbability. Engr. Jeffrey T. Dellosa.
Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationNovember 8, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol
More informationName: Section: Date:
WORKSHEET 5: PROBABILITY Name: Section: Date: Answer the following problems and show computations on the blank spaces provided. 1. In a class there are 14 boys and 16 girls. What is the probability of
More informationAdvanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY
Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY 1. Jack and Jill do not like washing dishes. They decide to use a random method to select whose turn it is. They put some red and blue
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationDiscrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
More informationCounting. Chapter 6. With Question/Answer Animations
. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Counting Chapter
More informationCourse Learning Outcomes for Unit V
UNIT V STUDY GUIDE Counting Reading Assignment See information below. Key Terms 1. Combination 2. Fundamental counting principle 3. Listing 4. Permutation 5. Tree diagrams Course Learning Outcomes for
More informationThis Probability Packet Belongs to:
This Probability Packet Belongs to: 1 2 Station #1: M & M s 1. What is the sample space of your bag of M&M s? 2. Find the theoretical probability of the M&M s in your bag. Then, place the candy back into
More informationPermutations and Combinations
Permutations and Combinations Rosen, Chapter 5.3 Motivating question In a family of 3, how many ways can we arrange the members of the family in a line for a photograph? 1 Permutations A permutation of
More informationCS100: DISCRETE STRUCTURES. Lecture 8 Counting - CH6
CS100: DISCRETE STRUCTURES Lecture 8 Counting - CH6 Lecture Overview 2 6.1 The Basics of Counting: THE PRODUCT RULE THE SUM RULE THE SUBTRACTION RULE THE DIVISION RULE 6.2 The Pigeonhole Principle. 6.3
More informationW = {Carrie (U)nderwood, Kelly (C)larkson, Chris (D)aughtry, Fantasia (B)arrino, and Clay (A)iken}
UNIT V STUDY GUIDE Counting Course Learning Outcomes for Unit V Upon completion of this unit, students should be able to: 1. Apply mathematical principles used in real-world situations. 1.1 Draw tree diagrams
More informationLAMC Junior Circle February 3, Oleg Gleizer. Warm-up
LAMC Junior Circle February 3, 2013 Oleg Gleizer oleg1140@gmail.com Warm-up Problem 1 Compute the following. 2 3 ( 4) + 6 2 Problem 2 Can the value of a fraction increase, if we add one to the numerator
More informationUnit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22
Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/2011 1 / 22 Chapter 13: What is a probability Denition The probability that an event happens is the percentage
More informationStatistics Intermediate Probability
Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting
More informationCOUNTING AND PROBABILITY
CHAPTER 9 COUNTING AND PROBABILITY Copyright Cengage Learning. All rights reserved. SECTION 9.2 Possibility Trees and the Multiplication Rule Copyright Cengage Learning. All rights reserved. Possibility
More informationUnit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22
Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/2011 1 / 22 Chapter 13: What is a probability Denition The probability that an event happens is the percentage
More information10-1. Combinations. Vocabulary. Lesson. Mental Math. able to compute the number of subsets of size r.
Chapter 10 Lesson 10-1 Combinations BIG IDEA With a set of n elements, it is often useful to be able to compute the number of subsets of size r Vocabulary combination number of combinations of n things
More informationContent. 1 Understanding and analyzing algorithms. 2 Using graphs and graph algorithms
Content 1 Understanding and analyzing algorithms 2 Using graphs and graph algorithms 3 Using combinatorial reasoning and probability to quantitatively analyze algorithms and systems 3.1 Basics of Counting:
More informationMath 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
More informationPROBABILITY Case of cards
WORKSHEET NO--1 PROBABILITY Case of cards WORKSHEET NO--2 Case of two die Case of coins WORKSHEET NO--3 1) Fill in the blanks: A. The probability of an impossible event is B. The probability of a sure
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More informationCS 237 Fall 2018, Homework SOLUTION
0//08 hw03.solution.lenka CS 37 Fall 08, Homework 03 -- SOLUTION Due date: PDF file due Thursday September 7th @ :59PM (0% off if up to 4 hours late) in GradeScope General Instructions Please complete
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationJong C. Park Computer Science Division, KAIST
Jong C. Park Computer Science Division, KAIST Today s Topics Basic Principles Permutations and Combinations Algorithms for Generating Permutations Generalized Permutations and Combinations Binomial Coefficients
More informationMath 3201 Unit 3: Probability Name:
Multiple Choice Math 3201 Unit 3: Probability Name: 1. Given the following probabilities, which event is most likely to occur? A. P(A) = 0.2 B. P(B) = C. P(C) = 0.3 D. P(D) = 2. Three events, A, B, and
More informationProbability Paradoxes
Probability Paradoxes Washington University Math Circle February 20, 2011 1 Introduction We re all familiar with the idea of probability, even if we haven t studied it. That is what makes probability so
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting
More informationaabb abab abba baab baba bbaa permutations of these. But, there is a lot of duplicity in this list, each distinct word (such as 6! 3!2!1!
Introduction to COMBINATORICS In how many ways (permutations) can we arrange n distinct objects in a row?answer: n (n ) (n )... def. = n! EXAMPLE (permuting objects): What is the number of different permutations
More informationProbability. Dr. Zhang Fordham Univ.
Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!
More informationCS1800: Intro to Probability. Professor Kevin Gold
CS1800: Intro to Probability Professor Kevin Gold Probability Deals Rationally With an Uncertain World Using probabilities is the only rational way to deal with uncertainty De Finetti: If you disagree,
More informationCHAPTER 8 Additional Probability Topics
CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information
More informationSection The Multiplication Principle and Permutations
Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
More informationGrade 6 Math Circles Fall Oct 14/15 Probability
1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.
More informationCSE 312 Midterm Exam May 7, 2014
Name: CSE 312 Midterm Exam May 7, 2014 Instructions: You have 50 minutes to complete the exam. Feel free to ask for clarification if something is unclear. Please do not turn the page until you are instructed
More informationPermutations and Combinations Section
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Permutations and Combinations Section 13.3-13.4 Dr. John Ehrke Department of Mathematics Fall 2012 Permutations A permutation
More information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More informationPoker Hands. Christopher Hayes
Poker Hands Christopher Hayes Poker Hands The normal playing card deck of 52 cards is called the French deck. The French deck actually came from Egypt in the 1300 s and was already present in the Middle
More informationDate. Probability. Chapter
Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games
More informationBayes stuff Red Cross and Blood Example
Bayes stuff Red Cross and Blood Example 42% of the workers at Motor Works are female, while 67% of the workers at City Bank are female. If one of these companies is selected at random (assume a 50-50 chance
More informationMat 344F challenge set #2 Solutions
Mat 344F challenge set #2 Solutions. Put two balls into box, one ball into box 2 and three balls into box 3. The remaining 4 balls can now be distributed in any way among the three remaining boxes. This
More informationMATH 1324 (Finite Mathematics or Business Math I) Lecture Notes Author / Copyright: Kevin Pinegar
MATH 1324 Module 4 Notes: Sets, Counting and Probability 4.2 Basic Counting Techniques: Addition and Multiplication Principles What is probability? In layman s terms it is the act of assigning numerical
More informationProbability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37
Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete
More informationMath 14 Lecture Notes Ch. 3.3
3.3 Two Basic Rules of Probability If we want to know the probability of drawing a 2 on the first card and a 3 on the 2 nd card from a standard 52-card deck, the diagram would be very large and tedious
More informationMore Probability: Poker Hands and some issues in Counting
More Probability: Poker Hands and some issues in Counting Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the
More informationSTAT Statistics I Midterm Exam One. Good Luck!
STAT 515 - Statistics I Midterm Exam One Name: Instruction: You can use a calculator that has no connection to the Internet. Books, notes, cellphones, and computers are NOT allowed in the test. There are
More informationHere are two situations involving chance:
Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)
More informationCS 237: Probability in Computing
CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 5: o Independence reviewed; Bayes' Rule o Counting principles and combinatorics; o Counting considered
More informationMultiple Choice Questions for Review
Review Questions Multiple Choice Questions for Review 1. Suppose there are 12 students, among whom are three students, M, B, C (a Math Major, a Biology Major, a Computer Science Major. We want to send
More informationINDIAN STATISTICAL INSTITUTE
INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 20-07-07 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 19 February 2014 Poker II 19 February 2014 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationVenn Diagram Problems
Venn Diagram Problems 1. In a mums & toddlers group, 15 mums have a daughter, 12 mums have a son. a) Julia says 15 + 12 = 27 so there must be 27 mums altogether. Explain why she could be wrong: b) There
More informationIndependent and Mutually Exclusive Events
Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A
More informationIn this section, we will learn to. 1. Use the Multiplication Principle for Events. Cheesecake Factory. Outback Steakhouse. P.F. Chang s.
Section 10.6 Permutations and Combinations 10-1 10.6 Permutations and Combinations In this section, we will learn to 1. Use the Multiplication Principle for Events. 2. Solve permutation problems. 3. Solve
More informationName Instructor: Uli Walther
Name Instructor: Uli Walther Math 416 Fall 2016 Practice Exam Questions You are not allowed to use books or notes. Calculators are permitted. Full credit is given for complete correct solutions. Please
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
More informationIntroductory Probability
Introductory Probability Combinations Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Assigning Objects to Identical Positions Denitions Committee Card Hands Coin Toss Counts
More informationPROBABILITY. 1. Introduction. Candidates should able to:
PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation
More information{ a, b }, { a, c }, { b, c }
12 d.) 0(5.5) c.) 0(5,0) h.) 0(7,1) a.) 0(6,3) 3.) Simplify the following combinations. PROBLEMS: C(n,k)= the number of combinations of n distinct objects taken k at a time is COMBINATION RULE It can easily
More informationSTAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes
STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle
More informationChapter 1: Sets and Probability
Chapter 1: Sets and Probability Section 1.3-1.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing
More informationActivity 1: Play comparison games involving fractions, decimals and/or integers.
Students will be able to: Lesson Fractions, Decimals, Percents and Integers. Play comparison games involving fractions, decimals and/or integers,. Complete percent increase and decrease problems, and.
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More information6/24/14. The Poker Manipulation. The Counting Principle. MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical experiments
The Poker Manipulation Unit 5 Probability 6/24/14 Algebra 1 Ins1tute 1 6/24/14 Algebra 1 Ins1tute 2 MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability models MAFS. 7.SP.3:
More information