L. #70 How many ways can it happen? Ordering Items

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1 L. #70 How many ways can it happen? Ordering Items A2.S.10 A2.S.12 Calculate the number of possible permutations (npr) of n items taken r at a time Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) To begin our study of probability we will learn how to find the number of ways an event can happen. Fundamental Counting Principle: If there are m ways to do one thing, and n ways to do another, then there are m n ways of doing both. The Fundamental Counting Principle is the guiding rule for finding the number of ways to accomplish two or more tasks. Ex 1) For example, if a menu offers (A) four different beverages and (B) five different main dishes there are different meals. Ex 2) If you want to draw 2 cards from a standard deck of 2 cards without replacing them, then there are 2 ways to draw the first and 1 ways to draw the second, so there are a total of 2 1= 262 ways to draw the two cards. n(e) means number of events in E. The notation is similar to function notation. We will call choosing an outfit event E. is choosing a shirt, is choosing a pair of pants, is choosing a pair of shoes, and is choosing a jacket. Given your current wardrobe, how many different outfits could you make? (Estimate the number of each item in your wardrobe.) N( )= N( )= N( )= N( )= N(E)= Exercise #1: Ten runners are entering in the 100-yard dash. How many ways could the runners finish first, second, and third? Exercise #2: If you had twelve songs on your mp player, and you set the play mode to random, how many different ways could you listen to the 12 songs? ~ 1 ~

2 Permutations and Combinations on the calculator a) Enter the n value b) MATH PRB #2 or # c) Enter the r value A. How many different Saugerties telephone numbers (26 and 27) can be created? B. License plates typically have 6 characters. There are 26 letters and 10 to choose from. How many different license plates can be made if no digits or letters can be repeated and it must begin with four letters followed by two numbers. Remember, permutations are really just a special application of the counting principle. In any situation, if can use a permutation, then you could also use the counting principle. (The converse is not true. There are situations where you use the counting principle but cannot use permutations. Look at A & B.) ~ 2 ~

3 C. Sienna has 6 blocks with one of the letters A, B, C, D, E, and F on each one. 1) In how many ways can she arrange the letters? 2) In how many ways can she arrange the letters if the first and last must be vowels? Permutations with Repetition 1. How many ways can you arrange the letters in the word: POINT? Write out the product for P. 2. Write down the arrangements you can make with the letters in the word, SEE. How could you rewrite this product using factorial notation? In general any permuation in the form P n! n n Normally, you would think there would be P! 6 different ways to arrange the letters. Because of the repeated E s, some of the arrangements will be the same. Since there are 2 E s there are 2 P 2 2! repeats. Therefore the total number of arrangements of the letters in the word SEE is!. 2! Let s consider the word, RIFFRAFF. There would be P P 8 8 P 2 2 8!!2! 80 ways to arrange the letters. The 8! represents all of the letters, the! represents the repeated F s, and the 2! represents the repeated R s.. How many ways can you arrange the letters in the word: KAYAK?. How many ways can you arrange the letters in the word: ELEVEN?. How many ways can you arrange the letters in the word: DODECAHEDRON? 6. How many ways can you arrange the letters in the word: MATHEMATICS? ~ ~

4 L. #71 How many ways can it happen? Grouping Items A2.S.9 Differentiate between situations requiring permutations and those requiring combinations A2.S.11 Calculate the number of possible combinations (ncr) of n items taken r at a time A2.S.12 Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) In each of the problems on the first two pages, we were ordering or arranging the objects. For example, in the race we were not just trying to choose the top three, we wanted to specify who came in first, second, and third. When the order does not matter, if we did just want to choose the top three in general, we use a different measurement called a combination. n is another notation r C for n r. For example, C. In essence, a combination is similar to a permutation, but the order of the items does not matter. Therefore, if we started with a permutation, we would remove all of the situations where there are the same items in the group but in a different order. Lets look at a simple example to see how this works. Consider the letters in the word MATH. Below are all of the different ways that we can arrange three of these letters. MAT MTA AMT ATM TMA ATM MAH MHA AMH AHM HMA HAM MTH MHT TMH THM HMT HTM -Use a permutation to make sure that I did not miss any of the possible arrangements. ATH AHT TAH THA HAT HTA -Now lets assume that the order of the letters does not matter. In other words, we just want to group three letters together. Cross out any of the groups above that are repeats. How many groups are you left with? -Try C in your calculator to confirm your answer. -In is the combination formula, n C r n P r r!, what does r! do? Exercise #1: Ten runners are entering in the 100-yard dash. How many different groups of three people could win the top three prizes? ~ ~

5 1. In each situation determine if you would use a permuation or a combination. a.) Choosing basketball starters. b.) Choosing a president, vice president, secretary, and treasurer of a club. c.) Making a password using random numbers. d.) Selecting a group of marbles from a jar. 2. There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send four representatives to the State Conference. a.) How many different ways are there to select a group of four students to attend the conference? b.) If the members of the club decide to send two juniors and two seniors, how many different groupings are possible? ~ ~

6 c.) If the members of the club decides to send one junior and three seniors, how many different groupings are possible? d.) If the four chosen members are announced at the conference, in how many different ways can their names be read?. If a three digit number is formed from the digits 1,2,,,,6, and 7, with no repetitions, tell how many of these three digit numbers will have a number value between 200 and 00?. Adrianna is trying to choose DVD s to take with her on vacation. She is selecting from six action movies, eight comedies, and four dramas. a.) How many sets of DVD s could she select? b.) How many sets of DVD s would contain all comedies? c.) How many sets of DVD s would contain two action movies, two comedies, and one drama? d.) Once she chooses the DVD s, in how many different orders can she watch them?. How many ways can you arrange the letters in the word: JENNIFER? ~ 6 ~

7 Lesson #72 Theoretical and Empirical Probability A2.S.1 Calculate theoretical probabilities, including geometric applications A2.S.1 Calculate empirical probabilities Probability is the area of mathematics that answers the question, what is chance that an event will occur? It can be expressed as a fraction, a decimal, or a percent. The general formula for the probability of a single event is as follows: For example, if you have a bag of marbles of which are red, 1 is blue, and 6 are yellow, the theoretical probability of getting a red marble is: n( red) P(red). This probability could be expressed as. or 0%. n(total marbles) 10 We call this probability theoretical because that is what we expect to happen. Theoretically, out of 10 times (or 0% of the time) we should choose a red marble. Is that always what happens? Do you always get heads when you flip a coin 0% of the time? Do you always get a when you roll a dice 1/6 th of the time? The actual probability you get when you do an experiment is called the empirical probability. This makes sense because empirical means information gained by means of observation, experience, or experiment. How do you think the number of trials, (number of times a person picks a marble) impacts the relationship between the similarity between the theoretical probability and the empirical probability? Law of Large Numbers - as the number of trials of an experiment increases, the empirical probability approaches the theoretical probability. A. Think about the marble experiment. What is the theoretical probability you will choose a purple marble? B. What is the theoretical probability you will choose a marble and not a stone? C. What is the theoretical probability you will choose a marble that is NOT red? ~ 7 ~

8 From the three questions on the previous page we learn two important principles: A. Range for Probability: B. Probability of the Complement of E (opposite of an event): (notation E or E c 1 The probability and its complement complete each other because their sum is 1 or 100%. E c E ). 1. The party registration of the voters in Jonesville is shown in the table. If one of the registered Jonesville voters is selected at random, what is the probability that the person selected in not a Democrat? Party # of voters registered Democrat 6,000 Republican,00 Independent, Geologists say that the probability of a major earthquake occurring the San Francisco Bay area in the next 0 years is about 90%. Is this empirical probability or theoretical probability?. A fair die is tossed. The results appear in the table at the left. a) Based on this data, what is the empirical probability of tossing a?. What is the theoretical probability of tossing a? Result Frequency A standard deck of cards contains 2 cards. There are suits: hearts, diamonds, clubs and spades. Each suit contains 1 cards. You are asked to select a card from the deck without looking. b) What is the probability of drawing a ten? c) What is the probability of drawing a jack OR a diamond? ~ 8 ~

9 d) What is the probability that the card is red AND a king? e) What is the probability of drawing a black ace OR any face card? 6. A die is tossed. It is a fair, unbiased die. Find the theoretical probability of each event. f) A number less than or equal to appears. g) An odd number is tossed. h) A number greater than 2 AND even appears. i) A number less than three OR greater than appears. 7. A letter is chosen at random from a given word. Find the probability that the letter is a vowel if the word is: j) ALGEBRA k) PROBABILITY l) Explain how you would calculate the empirical probability of randomly choosing a vowel if each person in the class randomly chose a letter. P Probability and Geometry desired outcomes desired area total outcomes total area Review of area formulas Rectangle/Parallelogram: Triangle: Circle: 8. A 10 x 20 foot mural, depicted below, shows a triangularly shaped region at the bottom of the mural. Find the probability that a point selected at random will lie in this triangular region of the mural. ~ 9 ~

10 9. A square dartboard is represented on the accompanying diagram. The entire dartboard is the first quadrant from x = 0 to 6 and from y = 0 to 6. A triangular region on the dartboard is enclosed by the graphs of the equations y = 2, x = 6, and y = x. Find the probability that a dart that randomly hits the dartboard will land in the triangular region formed by the three lines. 10. The radius of the bullseye is 1 and the radius of the largest circle is. If you threw a dart, what is the probability of hitting the bullseye, given that the dart will always land within the boundary of the outer circle? 11. The accompanying figure is a square. The interior sections are formed using congruent squares. If this figure is used as a dart board, what is the probability that the dart will hit the shaded blue region? 12. Challenge: What is the probability of a dart landing in the one of the shaded circles if it always hits somewhere on the square target? Round to the nearest percent. ~ 10 ~

11 Lesson #7 Probability with Multiple Events A2.S.12 Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) Independent Events: An occurrences or outcomes that do not affect each other Example: Flipping a coin twice. The probability of heads on the 1 st flip is independent of the 2 nd flip. The probability of two or more independent events occuring in a row, one AND then the other, can be found by multiplying the individual probabilities. (This is similar to the fundamental counting principle). AND = multiply Mutually exclusive events: Two or more events that cannot occur at the same time Example: Choosing two red marbles OR choosing a red and a green marble. If you choose two marbles, these events cannot both happen at the same time. The probability of one mutually exclusive event OR another occurring can be found by adding their individual probabilities. OR = add 1. A bag of cookies contains 6 chocolate chip cookies, peanut butter cookies, and 1 oatmeal cookie. Brandon selects a cookie and eats it. Then he selects another. Find the probability that Brandon selected : a) 2 chocolate chip cookies (chocolate chip AND chocolate chip) b) A chocolate chip cookie followed by an oatmeal cookie c) 2 chocolate chip cookies OR 1 chocolate chip cookie followed by 1 oatmeal cookie d) Anything but 2 chocolate chip cookies 2. Two colored dice (one red, one white) are rolled. What is the probability of rolling "box cars" (two sixes)? ~ 11 ~

12 . Brianna is using the two spinners shown below to play her new board game. She spins the arrow on each spinner once. Brianna uses the first spinner to determine how many spaces to move. She uses the second spinner to determine whether her move from the first spinner will be forward or backward. a. Find the probability that Brianna will move fewer than four spaces backward. b. Find the probability that Brianna will move one space forward or four spaces backward.. A bag contains 12 red M&Ms, 12 blue M&Ms, and 12 green M&Ms. What is the probability of drawing two M&Ms of the same color in a row? (When the first M&M is drawn, it is looked at and eaten.) OR OR. A student council has seven officers, of which five are girls and two are boys. If two officers are chosen at random to attend a meeting with the principal, what is the probability that one girl and one boy are chosen? 6. Alex's wallet contains four $1 bills, three $ bills, and one $10 bill. If Alex randomly removes three bills without replacement, determine the probability that the bills will total $11. ~ 12 ~

13 Probability of choosing certain groups In example, the calculation was tricky because you had to consider either a boy being first or a girl being chosen first. In example 6, you had to consider all of the ways you could select the bills totaling $11. This would be even more difficult if you had to choose four or more items where the order of choosing did not matter. There is an easier way. We can use combinations to count the ways that the group of two student council members or the group of three bills can be chosen and divide these by the total number of groups in each case. Go back to examples & 6 and calculate the each probability using combinations. ~ 1 ~

14 group Remember, you can only use combinations in probability when the order does not matter. In other words, you can use it when selecting a group. ~ 1 ~

15 Lesson #7 Binomial Probability A2.S.1 Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most In a binomial experiment there are two outcomes, often referred to as "success" and "failure". If the probability of success is p, the probability of failure is 1 p. We usually use the variable q for 1 - p. For example, if the probability of success, p, is.2, the probability of failure, q, is.8. Examples of binomial experiments: flipping a coin -- heads is success, tails is failure p= q= rolling a die -- is success, anything else is failure p= q= voting -- votes for candidate A is success, anything else is failure determining eye color -- green eyes is success, anything else is failure spraying crops -- the insects are killed is success, anything else is failure Binomial probability problems have the following characteristics: There are a number of successive trials Each trial has two possible outcomes Each trial is independent of the other trials (the outcome of one trial does not impact the outcomes of the other trials.) They words exactly, at least, or at most appear in the problem, usually in italics (this is not always true, but the regents questions will be set up in this way) There is a moderately complex formula for calculating binomial probability, but this can also be done easily on the calculator if you can find three components of the problem: the number of trials (n), the probability of a single success (p), and the number of desired successes (r). First, we will work on identifying these components. n: number of trials p: probability of success on a single trial r: number of desired successes 1. The SHS baseball team has a chance of winning each game it plays this year. What is the probability it wins exactly three of its first five games? ~ 1 ~

16 2. The SHS baseball team has a chance of winning each game it plays this year. What is the probability it wins at most three of its first five games?. The SHS baseball team has a chance of winning each game it plays this year. What is the probability it wins at least three of its first five games?. On an equatorial island it rains very consistently. The probability of rain on any given day is 2/. What is the probability it will rain exactly five days next week?. If you drive on 9W through a section of Kingston, there are six traffic lights in a row. On average, each traffic light is red for 0 seconds, yellow for seconds, and green for 20 seconds. Find, to the nearest tenth, the probability that you would get a red light no more than two times. 6. A study shows that % of the fish caught in a local lake had high levels of mercury. Suppose that 10 fish were caught from this lake. Find, to the nearest tenth of a percent, the probability that at least 8 of the 10 fish caught contained high levels of mercury 7. On the SAT s there are five choices per question. On a section of 20 questions, what is the probability that someone could randomly guess correctly on at least 1 of the questions? You will notice that the p-value is usually given in the beginning of the problem while n and r values are given towards the end of the problem. This is not always the case, but it can help you decode the question. Once you can recognize a binomial probability problem and find n, p, and r, you can easily find the probability on your calculator. ~ 16 ~

17 Exact Binomial Probability on the Calculator 1. Go to 2 nd VARS (DISTR) 2. Arrow down and choose binompdf (This is either 0: or A: depending on your calculator.). Enter the information as follows: binompdf (number of trials(n), probability of success(p),number of successes(r)) Summary: binompdf (n,p,r) ~ 17 ~

18 To calculate binomial probability with at most/at least questions, the process is very similar. Since there are multiple r values, and you want one or the other of them, you want to add their probabilities together. You can tell your calculator to do so in the following way: At Most/At Least Binomial Probability on the Calculator 1. Press 2 nd STAT (LIST) MATH 2. Choose :Sum. Go to 2 nd VARS (DISTR). Arrow down and choose binompdf (This is either 0: or A: depending on your calculator.). Enter the information as follows: binompdf (number of trials(n), probability of success(p),{number of successes(r)}) Summary: sum(binompdf(n,p,{r}) The {} brackets around the r values indicates that all of those numbers are r values. sum( adds up the individual probability for each r value. 1. Sheffield is batting.267 for the Yankees. What is the probability that he will get at most 2 hits in his next 7 at bats? Round to the nearest thousandth. 2. The Saugerties wrestling team has won 7% of its matches over the past ten years. Assuming that their skill level is consistent each year, what is the probability that they will win at least of their last 7 matches? Round to the nearest percent.. If you drive on 9W through a section of Kingston, there are six traffic lights in a row. On average, each traffic light is red for 0 seconds, yellow for seconds, and green for 20 seconds. Find, to the nearest tenth, the probability that you would get a red light at most two times.. On the SAT s there are five choices per question. On a section of 20 questions, what is the probability that someone could randomly guess correctly on at least 1 of the questions? Express your answer as a decimal and a percent. ~ 18 ~ I can t find it! If you forget where to find sum or binompdf, all of the functions on your calculator can be located in the catalog. Press 2 nd 0 (Catalog) and scroll through the alphabetized options. You can press the button with first letter of the item you are looking for to get there faster. The letters are above the buttons in green.

19 Lesson #7 - Binomial Expansions A2.A.6 Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion In the following expressions, a and b could represent any single term. For example, would be the same as 1. Simplify (expand): ( a b) 2 ( a b ) where a=2x and b=. (2x ) 2. Simplify (expand): ( a b ) Here is the expansion of ( a b) a b a b a b a b a b a b a b a b a b 2 2 a ab ba b a b a b 2 2 a 2ab b a b a b a 2a b ab 2 2 ba 2ab b a a b ab b a b a a b a b ab 2 2 ba a b ab b 2 2 a a b 6a b ab b 2 2. How long do you think it would take you to find a b ( a b ) using that same method? In the problems above, you were performing a binomial expansion the long way, but a helpful pattern forms that allows you to perform the expansion of ( a b ) n without doing endless multiplication. This formula or pattern is known as the binomial theorem. ~ 19 ~

20 The formula for this pattern is actually given to you on the regents. Lets learn how to use it ( a b) C a b C a b C a b... C a b n n n n n n 0 n 1 n 2 n n Notice that a is the first term of the binomial, b is the second term of the binomial, and n is Example: Expand (2x ). the exponent on the expansion. 1. Identify n, a, and b. a = 2x b = n = 2. Plug those values into the formula. It helps to write the terms vertically. Simplify the resulting expression ( a b) n 0 C a b n 0 n 1 1 C a b n 1 n 2 2 C a b n 2... n n n 0 C a b n ( 2x ) C 0 C 1 C 2 C C 2x 2x 2 2x 1 2x 0 2x 2 Final Answer: 16x 160x + 600x x x x 8x 2 x 1 2 2x x 16x 160x 600x x+ 62 Perform each of the following binomial expansions: 1) (x 6) ~ 20 ~

21 2) (x 2 y) You will also be asked to identify specific terms in an expansion. What is the middle term in the example problem? What is the 2 nd term in #1? When you do not already have the expansion written out and you only have to find one term, you do not want to write out the entire expansion. Instead, use the formula and the up, down, up pattern to find the term you want ( a b) C a b C a b C a b... C a b n n n n n n 0 n 1 n 2 n n Example) What is the th term in the expansion of x? 1. Identify n, a, and b: n = a = x b = 2. Write out the first term. n. From there, count up (0,1,2,) for the combination, down (,,,2) for the exponent on a, and up (0,1,2,) for the exponent on b.. Simplify the resulting term. 1st term: C a b C x n th term: C x (10)( x )(6) 60x What is the middle term of 6 ( x y ). ~ 21 ~

22 2. What is the rd term of the expansion of 2i?. What is the last term of the expansion of( x 2 y) 2.. What is the th term of the expansion of 2 8 ( x y ). ~ 22 ~

23 Lesson #76 Summations A2.N.10 Know and apply sigma notation Our first quote of the year was, The purpose of mathematics is not to make simple things complicated, but to make complicated things simple. You have seen many examples of this fact throughout the year (multiplication for repeated addition, exponents for repeated multiplication, function notation, the unit circle, etc.). Now we will learn a way to express long sums using a compact notation: A summation is the sum of a group of numbers or expressions. For example, the sum of the consecutive integers 2 through 7 is.this sum is easy to write with words. Certain problems such as, the sum of n-1 where n is all integers from 2 through, are more easily understood written in sigma notation. The greek letter, sigma: abbreviate the summation of a group of numbers. n 2 n 1 is used in math to The letter below sigma identifies the index variable in the expression. This is the variable that we will plug the numbers into. Any other variables in the problem remain as variables. The number below sigma gives for the index variable. The number above sigma gives for the index variable. Altogether, this expression tells us to evaluate n-1 for all integers 2 through, and add the resulting values. Practice: 1. a 1 a 2 2a ~ 2 ~

24 2. x 1 sin x 2 Mode? 7 2. x x i i is the imaginary unit.. Find the value of i 0 i x 2y i i is the index variable 6. Find the value of 2 7 ( 1) k 0 k. PEMDAS 7. Evaluate: x 2 cos x 2 8. If n C r represents the number of combinations of n items taken r at a time, what is the value of C? 2 r r=1 (1) 8 () 12 (2) 28 () 8 ~ 2 ~

25 At the beginning of the lesson, we started with the reason for summation notation: to write long sums concisely. The binomial theorem that we learned in the last lesson is a good example of a long sum. It is also given to you in summation notation on the reference sheet ( a b) C a b C a b C a b... C a b n n n n n n 0 n 1 n 2 n n n n n r r a b C a b n r r 0 Since summations are so new to you, we will not be using the binomial theorem in this form. I have found that it only confuses students more at this point because there are so many variables within the summation notation. I only mention it now so that you can see how much more concise it is to write it this way. Summations on the Calculator These problems can be checked on the calculator as long as the problem does not contain any other variables other than the index variable. That means all of the problems in this lesson can be checked this way except for example. To do so, press ALPHA WINDOW (F2). Choose the second option, and enter the problem as you see it. Go back and try a couple of the example problems. You still must show your work when doing these problems by writing out the summation. Having the calculator should help you avoid careless mistakes. ~ 2 ~