Experiment: Outcome: Sample Space: Fair Unbiased Experiment: Probability: Odds: Relative Frequency: Observed Probability: Mutually Exclusive Events:

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2 Definitions Experiment: A situation that has several possible results. Outcome: The result of an experiment. Sample Space: The set of all possible outcomes. Fair or Unbiased Experiment: one in which all outcomes are equally likely. Probability: number of successes number of possible outcomes Odds: number of successes : number of failures Relative Frequency: The fraction of times an event has occurred in the past. Observed Probability: Probability based on past performance (relative frequency). Mutually Exclusive Events: Events that have no outcomes in common. Complimentary Events: Mutually exclusive events whose union is the entire sample space. Independent Events: Events such that the occurrence of one does not affect the probability of the occurrence of the other.

3 Sample Space A family has a new baby. Before the birth, they chose not to learn the gender. List the sample space. Boy, Girl A jar contains 8 blue, 10 red, 7 green, and 9 yellow marbles. A marble is selected blindly. List the sample space. Blue, Red, Green, and Yellow A quarter is flipped twice. List the sample space. HH, HT, TH, TT

4 Probability The probability of an event occurring, as stated earlier, is the number of possible outcomes in the sample space that satisfy the condition, divided by the total number of outcomes of the experiment. number of successes P(E) number of possible outcomes Probability is commonly written in any form of the value: fraction, decimal, or percent.

5 Probability A quarter is flipped twice. What is the probability both are heads? The outcomes are HH, HT, TH, TT. Only one of four contains both heads. What is the probability of at least one heads? A family has three children. If a boy and girl are considered equally likely, what is the probability all the children are girls? BBB, BBG, BGB, BGG, GBB, GBG, 1 GGB, GGG

6 Relative Frequency A family has three children. If a boy and girl are considered equally likely According to statistics, it appears boys and girls are not equally likely. The ratio of boy to girl births worldwide seems to be somewhere between 1.04 and Many of the discussed principles of probability are null and void if outcomes are not equally likely. The example with the coin is another instance. If heads and tails are not equally likely, our work on the last example is flawed.

7 Relative Frequency All this gives rise to relative frequency. Relative frequency is a measure of how often something has occurred in the past. It s then assumed the next trial has the same probability. Consider tossing a thumbtack into the air. It can land point down or point up. Are the two outcomes equally likely? I doubt it, so the probability of it landing point up is unlikely to be exactly 0.5. We d need to perform many, many trials to determine relative frequency.

8 Relative Frequency A weather forecast is based on relative frequency. If it s reported the probability of rain today is 20%, it means that under these weather conditions barometric pressure, temperature, wind direction and speed, moving fronts, etc., it has rained 20% of the time in the past. Relative frequency 0 means the event has never happened, but probability 0 means it s impossible. Relative frequency 1 means the event has occurred in all trials to date. Probability 1 means the event must occur.

9 Probability A jar contains 8 blue, 10 red, 7 green, and 9 yellow marbles. A marble is selected blindly. What is the probability it is a blue marble? What is the probability it is not yellow? What is the probability the marble is black? 0 What is the probability it is not black? 1

10 Probability A jar contains 8 blue, 10 red, 7 green, and 9 yellow marbles. A marble is selected blindly. What is the probability it is not a yellow marble? You probably used this method: Here s another approach. The probability of a yellow marble and not a yellow marble are complimentary events. They can t both occur (mutually exclusive), yet one must occur (the sum of their probabilities is 1).

11 Properties of Probability A jar contains 8 blue, 10 red, 7 green, and 9 yellow marbles. A marble is selected blindly. What is the probability it is not a yellow marble? P(not yellow) = 1 P(yellow) = Properties of Probability 1) 0 < P(E) < 1 2) If E is an impossible event, P(E) = 0 3) If E is a sure thing, P(E) = 1 4) If E and F are complimentary events, P(E) 1 P(F) 25 34

12 Probability with Dice When rolling a die, find: P(4) P(even) P(0) P(less than 7) When rolling a pair of dice, find: P(a sum of 6) P(a sum greater than 9) P(a sum greater than 15) P(a sum less than 15)

13 Probability with Dice When rolling a pair of dice, find: 5 P(a sum of 6) 36 P(a sum greater than 9) P(a sum greater than 15) P(a sum less than 15)

14 Odds A jar contains 8 blue, 10 red, 7 green, and 9 yellow marbles. A marble is selected blindly. What are the odds it is a yellow marble? Odds are the ratio of successes to failures. The odds of drawing a yellow marble are 9 : 25. The weatherman says the probability of rain today is 20%. What are the odds it will rain? The odds of rain are 20 : 80 = 1 : 4.

15 Probability A committee of Mark, Anna, Will, Stan and Betty pick two members randomly to form a subcommittee. Find: P(Mark and Stan are on the sub) P(Anna is on the subcommittee) P(Anna is not on the subcommittee) 5 3 MA MW MS MB AW AS AB WS WB SB 2 5

16 Standard Deck of Cards You need to be familiar with a standard deck of playing cards: 52 cards, no jokers, 4 suits of 13 cards each, 3 face cards in each suit.

17 Probability with Cards What is the probability of getting a king or an ace in one random draw from a standard deck of playing cards?

18 Probability with Cards What is the probability of getting a king or an ace in one random draw from a standard deck of playing cards? U is a set operation symbol read, union. It means to combine, or put together, the set elements. It corresponds to or. If two finite sets A and B are mutually exclusive, N(A U B) = N(A) + N(B). If A and B are mutually exclusive events, P(A U B) = P(A) + P(B).

19 Probability with Cards What is the probability of getting a king or an ace in one random draw from a standard deck of playing cards? Drawing a king and ace are mutually exclusive events. In other words, they can t both occur in the same draw. N(K U A) = N(K) + N(A) = = 8. P(K U A) = P(K) + P(A) = 4 / / 52 = 2 / 13

20 Probability with Cards What is the probability of getting a king or a heart in one random draw from a standard deck of playing cards?

21 Probability with Cards What is the probability of getting a king or a heart in one random draw from a standard deck of playing cards? Drawing a king and drawing a heart are not mutually exclusive events because you can draw the king of hearts. If you just add, you ve counted the king of hearts twice. Addition Counting Principle N(A U B) = N(A) + N(B) N(A B) N(K U H) = N(K) + N(H) N(K H) = = 16. P(K U H) = P(K) + P(H) - P(K H) = 4 / / 52 1 / 52 = 4 / 13

22 Mutually Exclusive Are the following pairs of events mutually exclusive? Earning an A or a B in this class yes Placing 1 st, 2 nd, or 3 rd in a tournament yes Roll an even number or one that s prime with a pair of dice. no The same number is showing on each of a pair of dice or the sum is odd. yes The same number is showing on each of a pair of dice or the sum is even. no Draw a red card or a face card from a standard deck of playing cards. no

23 Disjoint sets have no intersection. They share no elements that are a success. (ex: A = King and B = Ace) The elements shared by A and B in A B are a success for both conditions. (ex: King of Hearts) Counting using Sets In computing probability, you ll need to count successes. Sets can often help. The sets below represent A U B (A or B)

24 Counting using Sets 13 of the 50 states include territory west of the Continental Divide. 42 states include territory east of the Continental Divide. Is this possible? It is because some states have territory on both sides. How many states? 5 states N(W U E) = N(W) + N(E) N(W E) 50 = N(W E) N(W E) = 5

25 Counting using Sets 13 of the 50 states include territory west of the Continental Divide. 42 states include territory east of the Continental Divide. Is this possible? Five states have territory on both sides of the Continental Divide. Name them. Montana, Wyoming, Colorado, New Mexico, Alaska

26 Counting using Sets The 2007 graduating class at Scotus contained 62 students. 19 were enrolled in Calculus 2 nd semester and 21 were enrolled in Physics. 11 took both classes. How many seniors took at least one of the classes? How many took neither? At least one: 29 Neither: 33

27 Counting using Sets The 2007 graduating class at Scotus contained 62 students. 19 were enrolled in Calculus 2 nd semester and 21 were enrolled in Physics. 11 took both classes. How many seniors took at least one of the classes? How many took neither? N(C U P) = N(C) + N(P) N(C P) N(C U P) = = 29, etc. P(A) = 0.4, P(B) = 0.35, P(A and B) = 0.15 Find P(A or B). P(A U B) = P(A) + P(B) P(A B) P(A or B) = = 0.6

28 Probabililty with Dice In the casino dice game craps, the roller wins if he/she rolls a sum of 7 or 11 with a pair of dice on the first roll. What is the probability of winning on the first roll? Rolling a 7 and rolling an 11 with a pair of dice are mutually exclusive events. N(7 U 11) = N(7) + N(11) N(7 11) = = 8 P(7 U 11) = 2 / 9 P(7 U 11) = P(7) + P(11) = 6 / / 36

29 Tree Diagrams The last example, those with sets, and pair of dice examples point out the value of diagrams, sketches, etc. Here s another. A Mexican restaurant serves four specialty dishes: enchiladas, fajitas, burrito supreme, and a taco dinner. The customer can then choose a side of beans, rice, or nacho chips. How many different menu choices are available? 12

30 Tree Diagrams A Mexican restaurant serves four specialty dishes: enchiladas, fajitas, burrito supreme, and a taco dinner. The customer can then choose a side of beans, rice, or nacho chips. How many different menu choices are available? 12 Multiplication Counting Principle If A and B are finite sets, the number of ways to choose one element from A, then one element from B is N(A) N(B).

31 Tree Diagrams A Scotus student has 3 shirts they can wear to school, 2 pair of shoes, 2 pair of pants that meet dress code, and 3 pair of shorts. Not counting socks, belts, etc., and ruling out the addition of a sweatshirt, how many wardrobes does the student have to choose from?

32 Tree Diagrams A student has 3 shirts and 2 pair of shoes and 2 pair of pants or 3 pair of shorts. How many wardrobes does the student have to choose from? 3 2 (2 + 3) = 30 Generally in probability, and means multiply and or means add.

33 And and Or A family plans a western vacation. They have brochures to 6 ski resorts and 4 dude ranches. How many choices are possible with these destinations if they can afford only one ski resort or one dude ranch? = 10 How many choices are possible with these destinations if they can afford one ski resort and one dude ranch? 6 4 = 24 With probability, generally or means add, and means multiply.

34 Review Are any of the events listed to the right mutually exclusive? In one draw from a standard deck: A: Draw an even card. B: Draw a heart. C: Draw a face card. A and C are mutually exclusive. From the same events, find: N(A U C) P(A U C) N(B C) P(B C) P(B U C)

35 Strings with Replacement How many possible answer keys are there in a five-problem True-False quiz? We could write all the possibilities, like HH, HT, TH, TT when a coin is tossed twice, but when there are several trials, the Fill-the- Blank method is often helpful. There are two choices on the first problem and two choices on the next, etc. This implies multiplication Of course, this is 2 5 = 32

36 Strings with Replacement How many possible answer keys are there in a five-problem True-False quiz? = 2 5 = 32 T or F is similar to a pool of possibilities. Once T or F is selected, it is returned to the pool to potentially be chosen the next time. These are selections with replacement. Selection with Replacement Let S be a set with n elements. Then there are n k possible arrangements of k elements from S with replacement.

37 Strings with Replacement Return to the problem with the five True- False questions. Imagine you did not read the material. Each answer is a guess. What is the probability you score 100%? 1 3% 32 What is the probability you earn a passing score? You could get them all correct, or miss one problem. If you miss two problems, your score is 60% %

38 Independent and Dependent What is the probability of flipping three heads in a row? What is the probability of rolling three sevens in a row with a pair of dice? What is the probability of drawing three aces in a row from a standard deck, with replacement? What is the probability of drawing three aces in a row from a standard deck, without replacement?

39 Independent and Dependent The difference between the last two problems is an issue of independent and dependent events. When the ace is placed back in the deck, the probability of drawing it the next time is the same as the probability the time before. The probability of getting an ace on the second draw without replacement depends on the result of the first draw. You ve rolled three sixes in a row with a single die. What is the probability you ll roll 6 on the next roll? 1 6

40 Independent and Dependent You ve rolled three sixes in a row with a single die. What is the probability you ll roll 6 on the next roll? There are six items in the sample space, one of which is 6. It is equally as likely as the other outcomes, so the fourth roll is independent of the rolls before it. It is said, The die has no memory. Events A and B are independent events if and only if P(A B) = P(A) P(B). P(A B) = P(A) P(B, given A) if A and B are dependent events. 1 6

41 Independent and Dependent A drawer contains six black and six white socks, scattered about in no particular configuration. It is dark and you blindly draw two socks from the drawer. What is the probability the second sock you draw is black? The answer to this question depends on the color of the first sock because, without replacement, they are dependent events. What is the probability both socks are black?

42 Independent and Dependent A drawer contains six black and six white socks, scattered about in no particular configuration. It is dark and you blindly draw two socks from the drawer. What is the probability both socks are the same color? BB, BW, WB, WW. Two of four are the same, so the probability is 0.5, correct? NO 5 The events are not independent. P(BB) = as does P(WW). P(same) = P(BW) = as does P(WB) P(different) =

43 Independent and Dependent Find the following probabilities. P(draw 2 aces in succession with replacement) P(draw 2 aces in succession without replacement) P(choose 3 boys in a row without replacement from a class of 10 boys, 10 girls) P(roll an even number on a die and flip a heads on a coin) P(draw a blue, then red without replacement from a bag of 6 blue, 4 red, and 3 green marbles)

44 Independent and Dependent Events A and B are independent if and only if P(A B) = P(A) P(B). 25% of the players on a basketball team can dunk. 40% can hip sled 650 pounds or more. 4 of the 32 members of the team can dunk and hip sled 650 pounds or more. Are these independent events? 4 P(A B) P(A) P(B) = (.25)(.4) =.1 No

45 Independent and Dependent A four-sided (tetrahedral) die has faces numbered 1 through 4. The die is tossed twice. Are the following events independent? A: the first roll is 2 B: the sum of the rolls is even P(A) =.25 P(B) =.5 P(A) P(B) =.125 P(A B) =.125 C: the first roll is 3 D: the sum of the rolls is 6 P(C) =.25 P(D) =.1875 P(C) P(D) = P(C D) =.0625 Yes No

46 Independent and Dependent Player A is an 80% free throw shooter and Player B is a 75% free throw shooter. Each will shoot one free throw and results are independent of one another. Find: P(both make) P(neither make) (.8)(.75) =.6 (.2)(.25) =.05 P(exactly one makes) P(A make, B miss) + P(B make, A miss) = (.8)(.25) + (.75)(.2) =.35 P(at least one makes) P(one) + P(both) or 1 P(neither) =.95

47 Independent, mutually exclusive A and B are two events such that P(A) > 0 and P(B) > 0. P(A B) = P(A) P(B) if the events are independent. If P(A) and P(B) are both positive, P(A) P(B) is positive. For the events A and B to be mutually exclusive, there is no intersection and P(A B) = 0. P(A B) can t be both positive and 0, so non-impossible events can t be both mutually exclusive and independent. If you know events that could possibly occur are mutually exclusive, the are not independent. If you know they are independent, they are not mutually exclusive.

48 Strings w/o Replacement You ve narrowed your college search to five Catholic universities; Creighton, Marquette, Notre Dame, Seton Hall, and Gonzaga. You re ready to rank them from one to five. How many lists are possible? = 120 Another name for is 5!, read, five factorial. You have a factorial button on your calculator.

49 Strings w/o Replacement You ve narrowed your college search to Creighton, Marquette, Notre Dame, Seton Hall, and Gonzaga. You re ranking them from one to five. How many lists possible? = 120 The schools form a pool of possibilities. Once a school is selected, it is not returned to the pool. These are selections without replacement. Selection without Replacement There are n! permutations of n different elements without replacement.

50 Strings w/o Replacement The books are so large for Algebra II and Advanced Math that there are two teachers editions for each. We also have the gray Advanced Math book. How many ways can I arrange the five books on a shelf? 5! = 120 How many arrangements are there if I want to keep the Advanced Math and Algebra II books together? 3! 2!, but the Advanced Math or Algebra II books could be on the left, so 2(3! 2!) = 24

51 Permutations Permutations are ordered arrangements, meaning that the order of the elements is important. In the problem with the subcommittee, Mark and Stan was the same subcommittee as Stan and Mark. The order did not matter. That s an example of a combination, not a permutation. A class of 60 students elects a President, Vice-President, and Secretary-Treasurer. How many possibilities are there?

52 Permutations A class of 60 students elects a President, Vice-President, and Secretary-Treasurer. How many possibilities are there? Again, the Fill-the-Blank technique can work well here = 205,320 Since once the President is selected, they will not also be the Vice-President, etc., this is selection without replacement. The number of choices decreases with each blank.

53 Permutations 15 gymnasts compete. In how many ways can the gold, silver, and bronze medals be awarded? = 2730 This is called the permutation of 15 things taken 3 at a time, 3-item permutations formed from 15 selections. It is written 15 P 3. There are 10 horses in a race. How many possible trifectas are possible? 10 P 3 = 720

54 Permutations A few definitions regarding permutations n P r = n(n 1)(n 2) (n r + 1) For instance, 15 P 3 from the gymnasts problem was The last factor was ( ) = 13 n! n P r n r! For instance, 15 P 3 from the gymnasts problem would equal 15! 15! ! 15 3! 12! 12! Finally, 0! = 1 We ll see this at work soon.

55 Find 760! 758! Factorial These numbers are too large for your calculator to contend with, but it equals ! 758! = = 576,840

56 Permutations You are the manager of a 10-person slowpitch softball team. How many 10-player batting orders are possible with 10 players? 10! = 3,628,800 You ve made three decisions about your order. Your fastest player will bat first, your best power-hitter will bat fourth, and your pitcher will bat last. Now how many possible batting orders? _ 7! = 5040

57 Permutations Prove n P n = n! n! n! so n P n n r! n n! Prove 5 P r = 5 4P r-1 P r n 5! 5 r! 5 4! 4 r 1! n! 0! n! 1 n! 5! 4 r 1! 5! 5 r!

58 Permutations n P 5 = 20 n P 3. Find n. n! 20n! n - 5! n 3! n - 2 n - 3 n - 4 n n -1 n - 5! n - 5! 20n n -1 n - 3! n(n 1)(n 2)(n 3)(n 4) n - 2 n - 3! = 20n(n 1)(n 2) (n 3)(n 4) = 20 Expand and solve. n = 8 or n = -1. Can t have -1 P 5, so n = 8

59 Counting Number of Ways How many phone numbers are possible with a particular area code and prefix, for example; ? The answer is not , because once used, a digit can be used again. The choice for each blank is any digit = 10,000 More than one prefix is obviously necessary for Columbus, with population over 20,000. There are households that have only a cell number, but several households have more than one landline. The Columbus prefix also includes homes outside city limits.

60 Counting Number of Ways How many phone numbers are possible with Columbus prefixes 564, 563, and 562? 564- or 563- or , , ,000 30,000

61 Counting Number of Ways How many Platte County license plates are possible with 10-(a letter, and three digits)? = 26,000 There aren t this many, because several letters are not used. I believe that I, O, and Q are three letters you will not see. What about 10 - D5? There can be a letter followed by only one or two digits. We ve counted 10 D005. This could be considered the same as 10 D5. We ve counted 10 D000, however. How do you account for this? 26,000-26

62 Counting Number of Ways An exhibit hall has 8 doors. If you must enter and exit by different doors, how many possibilities are there? 8 7 = 56 What if there are no restrictions? 8 2 = 64 The 6 starters on a volleyball team have been determined. How many serving orders are possible? 6! = 720 How many starting basketball lineups can be created from 12 players on varsity? This is NOT 12 P 5 because order does not matter. This is 12 C 5. Use your calculator. = 792. More on combinations later. 12 C 5

63 Review All the letters in the alphabet are written on slips of paper and placed in a cup. Three are drawn at random and placed in the order they re drawn. Find P(BAT, CAT, HAT, MAT, PAT) P The sample space for an experiment is all ordered pairs (x,y). Let A be the event consisting of all (x,y) such that y = 3x. Name the event which is the compliment of A. All ordered pairs (x,y) such that y 3x

64 Review There are 6 different kinds of cereal boxes on the shelf. How many permutations are possible if Captain Crunch is placed on the far left, and Rice Krispies is on the far right? 4! = 24 Two boxes will be removed. How many four-box permutations are possible? 6 P 4 = 360

65 Review In a family of five, Mom or Dad always sit at the head of the table. In how many ways can they fill the five chairs? = 48 What is the probability that a 70% free throw shooter makes both of the two free throws they shoot? (.7) 2 = 49%. What is the probability the shooter makes exactly one? 2(.7)(.3) = 42%

66 Review Determine the number of permutations of the letters of the word ROCKS that begin with R. What is the probability of drawing two face cards in succession from a standard deck without replacement? What is the probability that the sum of a pair of dice is 6, or the first die thrown is a 6? What is the probability that the sum of a pair of dice is 7, or the first die thrown is a 6?

67 Review In a volleyball game between team A and B, are the events: Team A wins, Team B wins, mutually exclusive, complimentary, both, or neither? both In an election between candidates A, B, and C, are the events: A wins, B wins, C wins, mutually exclusive, independent, both, or neither? mutually exclusive

68 Review n P 7 = 12 n P 5. Find n. n! 12n! n 7! n 5! n 2 n 3 n 4 n 5 n - 6 n n -1 n - 7! n 7! 12n n -1 n 2 n 3 n 4 n 5! n 5! n(n 1)(n 2)(n 3)(n 4)(n 5)(n 6) = 12n(n 1)(n 2)(n 3)(n 4) (n 5)(n 6) = 12 Expand and solve. n = 9 or n = 2. Can t have 2 P 5 or 2 P 7, so n = 9

69 Review The ten guests at a party write their names on slips of paper and drop them into a hat for a drawing. You increase your chances of winning by writing your name on two slips of paper. Just before the drawing, you learn that a second name will be drawn for another prize. What is the probability that your name is drawn twice, thus revealing your misconduct? 1 55

70 Review Suppose that, though unknown to anyone, your friend Steve also had the idea of putting his name in the drawing twice. What is the probability that either you or Steve is exposed by the two draws? What is the probability that both you and Steve win a prize? or

71 Important Terms Review mutually exclusive: Events that can t both happen in the same trial. If events A and B are mutually exclusive, A B = Ø, so P(A B) = 0. complimentary events: Events that can t happen in the same trial, yet one must occur. If events A and B are complimentary, P(A) = 1 P(B) so P(A) + P(B) = 1. Independent events: Events such that the occurrence of one does not affect the probability of the occurrence of the other. Events A and B are independent iff P(A B) = P(A) P(B).

72 Review Definitions, Relationships, Random Thoughts number of successes P(E) number of possible outcomes Odds(E) = number of successes : number of failures With probability, generally and means multiply and or means add. P(A U B) = P(A) + P(B) P(A B) The number of ways to choose one element from A, then one element from B is N(A) N(B). If there are n elements in a set, there are n k possible arrangements of k elements with replacement. There are n! possible arrangements of all the elements without replacement.

73 Review Definitions, Relationships, Random Thoughts P(A B) = P(A) P(B given A) if A and B are dependent events. If events A and B have nonzero probability and are independent, P(A B) = P(A) P(B) > 0, so P(A B) 0. Therefore, independent events can t be mutually exclusive. where {(x 1,P(x 1 )), (x 2,P(x 2 )) (x n,p(x n ))} is the probability distribution., and 0! = 1.