1. AP Statistics: Section 8.2 Geometric Probability

2. Spence has trouble getting girls to say “Yes” when he asks them for a date. In fact, only 10% of the girls he asks actually agree to go out with him. Suppose that p = 0.10 is the probability that any randomly selected girl, assume independence, will agree to out with him. Spence desperately wants a date for the prom. (a) What is the probability that at least one of the first 5 girls asked will say “yes”. (b) How many girls can he expect to ask before the first one says “yes”?

3. In (b) we will let X = the number of times Spence needs to ask a girl for a date before a girl accepts. Why is this not a binomial distribution?

4. A random variable that counts the number of trials needed to obtain one success is called geometric and the distribution produced by this random variable is called a geometric distribution.

5. The Geometric Setting 1. Each observation falls into one of just two categories: _________ or _________. 2. The n observations are all _______________. 3. The probability of success, call it __, is __________ for each observation. 4. *The variable of interest is the number of trials required to obtain __________________.

6. Example 1: Consider rolling a single die. X = the number of rolls before a 3 occurs. Is this a geometric setting?

7. P(X = 1) = P(3 on 1 st roll) = P(X = 2) = P(not 3 on 1 st roll and 3 on 2 nd roll) = P(X = 3) = P(not 3 on 1 st or 2 nd roll and 3 on 3 rd roll) = P(X = 4) = P(not 3 on 1 st, 2 nd and 3 rd roll and 3 on 4 th roll) =

8. 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,.... 8If n is any one of these values, the probability that the first success occurs on the nth trial is:

9. TI83/84:

10. Example: What is the probability that the 6 th girl Spence asks to the prom will say “yes?”

11. Construct a probability distribution table for X = number of rolls of a die until a 3 occurs. Note that the number of table entries for X will be infinite. The probabilities are the terms of a geometric sequence, _______________, hence the name for this random variable. X: 1234567... P(X):

12. As with all probability distributions, the sum of the probabilities must be ___. Recall from Algebra II, maybe Pre-Calculus, that the sum of a geometric sequence is _________. So…

13. In the probability histogram, the first bar represents the probability of ________. The height of all subsequent bars is smaller since you are multiplying by a number less than 1. So the histogram will be _____-skewed. Always.

14. The Mean and Standard Deviation of the Geometric Random Variable If X is a geometric random variable with probability of success p on each trial, then

15. Example 2: A game of chance at the state fair involves tossing a coin into a saucer. You win a stuffed animal if the coin lands in and stays on the saucer. A person wins on average 1 out of every 12 times she/he plays. What is the expected number of tosses for a win? What is the standard deviation?

16. P(X > n) The probability that it takes more than n trials to see the first success is ________ Example 3: What is the probability that it takes more than 12 tosses to win a stuffed animal?