AP Stats Chapter 8. What are the parameters of a binomial distribution? How can you abbreviate this information?

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1 AP Stats Chapter 8 8.1: The Binomial Distribution What are the conditions for a binomial setting? What is a binomial random variable? What are the possible values of a binomial random variable? What are the parameters of a binomial distribution? How can you abbreviate this information? Example 1: Dice, Cars, and Hoops Determine whether the random variables below have a binomial distribution. Justify your answer. (a) Roll a fair die 10 times and let X = the number of sixes. (b) Shoot a basketball 20 times from various distances on the court. Let Y = number of shots made. (c) Observe the next 100 cars that go by and let C = color. (d) Deal 10 cards from a shuffled deck and count the number X of red cards. There are 10 observations, and each gives a red or black card. A success is a red card.

2 In general, how can we calculate binomial probabilities? Is the formula on the formula sheet? Example 2: Children Each child born to a particular set of parents has a 25% chance of having type O blood. If the parents have 5 children, what is the probability that exactly 2 of them have type O blood? Example 3: Switches Suppose you work for a manufacturing company that just received a shipment of switches. The number of switches that fail inspection has a binomial distribution with n = 10 and p = 0.1. What is the probability that in the shipment no more than 1 switch fails? Example 4: Switches again Rework the previous example on your calculator. Example 5: Testing Suppose you are taking a 10 question multiple choice test. Each question has 5 choices. Let X = # of correct guesses. Calculate P(X = 4).

3 Example 6: Basketball A basketball player makes 75% of her free throws. In a game she shoots 12 free throws and makes 7 of them. Is this unusual? What is the probability of making at most 7 baskets? Example 7: Testing Suppose you take a 10 question multiple choice quiz. Each question has five choices. You have no idea how to answer any of them, so you just guess on all 10. a) What is the probability that you get 8 correct? b) What is the probability that you get no more than 5 correct? c) What is the probability that you get 3 or more correct? Example 8: Graduation rates Suppose there are 20 basketball players at a certain university. For each athlete, the chance that they will graduate is 80%. a) What is the probability that 6 graduate? b) What is the probability that not all 20 graduate? c) What is the probability that no more than 13 graduate? d) What is the probability that more than 13 graduate?

4 How can you calculate the mean and standard deviation of a binomial distribution? Are these formulas on the formula sheet? Example 9: Calculate and interpret the mean and standard deviation for the binomial random variable in example 3. Example 10: Calculate and interpret the mean and standard deviation for the binomial random variable in example 5. What happens as the number of observations or trials gets larger and larger? What must be true to use the Normal Approximation? What notation is used with the Normal Approximation?

5 Example 11: Poll Shortly after Sept. 11, 2001 a poll of 400 adults asked, Do you approve of President Bush s reaction to 9/11? Bush s national approval rating at the time was 92%. a) What is the random variable X in this problem? b) Calculate P( X 358) c) What is the expected number of people who approve of the president? What is the standard deviation? d) Can we use the Normal Approximation to answer part (b)? e) Use the Normal Approximation to answer part (b). 8.2 The Geometric Distribution What is a geometric random variable? What are the possible values of a geometric random variable? What are the conditions for a geometric setting? How do I calculate geometric probabilities?

6 Example 12: Monopoly In the board game Monopoly, one way to get out of jail is to roll doubles. Suppose that a player has to stay in jail until he or she rolls doubles. The probability of rolling doubles is 1/6. (a) Explain why this is a geometric setting. (b) Define the geometric random variable and state its distribution. (c) Find the probability that it takes exactly three rolls to get out of jail. Example 13: From past experience it is known that 3% of accounts in a large accounting population are in error. What is the probability that the 6th account audited is the first one found with an error? Example 14: Repeat the previous example with your calculator. Example 15: Hockey A top NHL hockey player scores on 93% of his shots in a shooting competition. What is the probability that the player will not miss the goal until his 20th try? Probability Histograms for a Geometric Distribution: Suppose we roll a die until we get a 3. Calculate the following: a) P(X=1) b) P(X=2) c) P(X=3) d) P(X=4) Use the probabilities above to create a histogram for a geometric distribution. What do you notice?

7 Example 16: Dice Suppose we roll a die until we get a 3. Calculate the following: a) P( X 4) b) P( X < 4) Example 17: Some biology students were checking the eye color for a large number of fruit flies. For an individual fly, suppose that the probability of white eyes is ¼, the probability of red eyes is ¾, and that we may treat these flies as independent trials. a) What is the probability that the first fly with white eyes is the fourth fly? b) What is the probability that at most four flies have to be checked for eye color to observe a white-eyed fly? c) What is the probability that at least four flies have to be checked for eye color to observe a white-eyed fly? In general, how do you calculate the mean and variance of a geometric distribution? Are the formulas on the formula sheet? Example 18: Monopoly On average, how many rolls should it take to escape jail in Monopoly? Example 19: If you are rolling a fair die, what is the expected number of rolls before a 1 comes out? Example 20: Go back to example 15. What is the expected number of shots before he misses?

8 Example 21: Roll a die until a 3 is observed. Calculate the following: a) What is the probability that it takes more than 6 rolls? b) What is the probability that it takes more than 12 rolls? c) P( X = 4) d) P( X 4) e) P( X < 4)