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AP Statistics: Section 6.3C Geometric Probability 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”? a ) B(5,.1) 1 - P(none) 1 - binomialpdf(5,.1,0) 1 - .59049 .40951 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? No fixed number of trials 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. The Geometric Setting 1. Each observation falls into one of just two categories: _________ failure success or _________. 2. The n observations are all _______________. independent 3. The probability of success, call it __, p is constant for each observation. __________ 4. *The variable of interest is the number of trials required to obtain __________________. the first success Example 1: Consider rolling a single die. X = the number of rolls before a 3 occurs. Is this a geometric setting? Yes P(X = 1) = P(3 on 1st 1 roll) = 6 P(X = 2) = P(not 3 on 1st roll and 3 on 2nd roll) = 5 1 5 6 6 36 P(X = 3) = P(not 3 on 1st or 2nd roll and 3 on 3rd roll) = 5 5 1 25 6 6 6 216 P(X = 4) = P(not 3 on 1st, 2nd and 3rd roll and 3 on 4th roll) = 5 5 5 1 125 6 6 6 6 1296 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: P(X n) (1 - p) n -1 p TI83/84: 2nd Vars D : geometpdf ENTER geometpdf(p, x) Example: What is the probability that the 6th girl Spence asks to the prom will say “yes?” geometpdf(.1,6) .059 Construct a probability distribution table for X = number of rolls of a die until a 3 occurs. X: P(X): 1 2 3 4 5 6 7 ... .1667 .1389 .1157 .0965 .0804 .0670 .0558 Note that the number of table entries for X will be infinite. The probabilities are the terms of a 2 3 geometric sequence, _______________, a, ar , ar , ar ,..... hence the name for this random variable. As with all probability distributions, the sum of the probabilities must be ___. 1 Recall from Algebra II, maybe Pre-Calculus, that a the sum of a geometric sequence is _________. 1 r So… p p 1 1 (1 p ) p In the probability histogram, the first bar represents the probability of ________. success 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. right 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 1 x p 1 p 2 p 2 x 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? 1 E( X ) x 12 1 12 11 1 1 12 12 132 11.489 2 1 1 144 12 P(X > n) The probability that it takes more than n trials to n ( 1 p) see the first success is ________ Example 3: What is the probability that it takes more than 12 tosses to win a stuffed animal? 12 1 P(X 12) (1 ) .3520 12