HW-Finish Practice (Attached)

HW-Finish Practice (attached) Ch. 8 Test THURSDAY 1-9-14 www.westex.org HS, Teacher Website 1-6-14 Warm up—AP Stats Mr. Lerner is a 60% free throw shooter. What is the chance that his first made free throw comes after his 5th shot? How is this question different from what we saw in section 8.1? Name ______Date ______AP Stats 8 The Binomial and Geometric Distributions 8.2 The Geometric Distributions

Objectives  Describe what is meant by a geometric setting.  Given the probability of success, p, calculate the probability of getting the first success on the nth trial.  Calculate the mean (expected value) and the variance of a geometric random variable.  Calculate the probability that it takes more than n trials to see the first success for a geometric random variable.  Use simulation to solve geometric problems.

Review of Binomial Random Variable With a binomial random variable, the # of trials is ______BEFOREHAND, and the binomial variable X counts the # of ______in that FIXED number of trials. If there are n trials, then the possible values of X are O, 1, 2, . . ., n.

New Situation There are situations in which the goal is to obtain a FIXED number of successes. If the goal is to obtain _____ success, a random variable X can be defined that counts the # of ______needed to obtain that FIRST success. A random variable that does this is called a ______random variable and the distribution it produces is a geometric distribution.

The possible values of a geometric random variable 1, 2, 3, . . . , (an infinite set) because it’s theoretically possible to proceed ______without ever obtaining a success.

The following situation are examples of a ______because they involve counting the # of trials until a ______occurs.

 Flip a coin until you get a tail.  Roll a die until you get a 5.  Attempt free throws until you hit one.

The Geometric Setting

1. Each observation falls into one of two categories, ______or ______. 2. The observations are all ______. 3. The probability of a success is the ______for each observation. 4. The variable of interest is the ______of trials required to obtain the ______success. Example--A game consists of rolling a single die. The event of interest is rolling a 5; this event is called success. The random variable is defined as X = the number of trials until a 5 occurs. Verify that this is a geometric setting.

1. Success or failure?

2. Observations independent?

3. Probability of success same for each event?

4. Variable of interest # of trials until success?

Using our last example let’s calculate the probability of X = 1, 2, 3, to see if we can discover a general rule for calculating Geometric Probabilities.

P(X = 1) =

P(X = 2) =

P(X = 3) =

Rule for Calculating Geometric Probabilities If X has a geometric distribution with probability p of success and (1 – p) of failure on each observation, the possible values of X are 1, 2, 3, . . .. If n is any one of these values, the probability that the first success occurs on the nth trial is: Let’s say Geena. has a 10 sided die and 9 of the sides have a 1 on them and the 10th side has a 2 on it. Assuming success is rolling a 2, what would you say is the expected number of times she’d have to roll the die to be successful?

Assuming success is rolling a 1, what is the expected number of times rolled to be successful?

The Mean (expected value ) of X for a Geometric Random Variable:

How about the variance of the two random variables above? The first random variable was X = # of trials until a 2 was rolled. The second random variable was Y = # of trials until a 1 was rolled. For which random variable would you expect there to be more variability in how long it takes to achieve success? Why do you feel the way you do?

The Variance of X for a Geometric Random Variable:

Look carefully at the variance formula above. Does it help to explain what we thought intuitively about the variability of each of our geometric random variables?

Mr. Lerner believes his probability of hitting a half court basketball shot at the West Essex Gym is 1 out of 10. Let X = the number of half court shots Mr. Lerner takes until he hits one. What is the probability that it takes him more than 10 shots to hit a half court shot?

Formula for P(X > n) The probability that it takes more than n trials to see the first success is:

YOU TRY: Roll a die until a 5 is observed. What is the probability that it takes more than 3 rolls to observe a 5? (look back at our example from the page before, notice anything?) Practice 1. Construct a probability distribution table for X = number of rolls of a die until a 5 occurs.

X 1 2 3 4 5 6 7 8 P(X)

2. Construct the probability histogram that corresponds with your table from #1. 8.45 Flip a Coin-Flip a coin until a head appears. a. Indentify the random variable X. b. Construct the pdf table for X. Then plot the probability histogram.

c. Add on to your pdf table with a row for cdf.

8.46 Arcade Game-Glenn likes the game at the state fair where you toss a coin into a saucer. You win if the coin comes to rest in the saucer without sliding off. Glen has played this game many times and has determined that on average he wins 1 out of every 12 times he plays. He believes that his chances of winning are the same for each toss. He has no reason to think that his tosses are not independent. Let X be the number of tosses until a win. Glenn believes that this describes a geometric setting. a. Use the formula for calculating P(X > n) to find the probability that it takes more than 10 tosses until Glenn wins a stuffed animal. b. Find the answer to a. by calculating the probability of the complement: 1 – P(X ≤ 10).

8.48 Language Skills-The State Department is trying to identify an individual who speaks Farsi to fill a foreign embassy position. They have determined that 4% of the applicant pool are fluent in Farsi. a. If applicants are contacted randomly, how many individuals can they expect to interview in order to find one who is fluent in Farsi?

b. What is the probability that they will have to interview more than 25 until they find one who speaks Farsi? More than 40?

8.49 Shooting free throws-A basketball player makes 80% of her free throws. We put her on the free-line and ask her to shoot free throws until she misses one. Let X = the number of free throws the player takes until she misses. a. What assumptions do you need to make in order for the geometric model to apply? What action constitutes “success” in this context?

b. What is the probability that the player will make 5 shots before she misses?

c. What is the probability that she will make at most 5 shots before she misses?