1. AP Statistics Chapter 8 Section 2

2. If you want to know the number of successes in a fixed number of trials, then we have a binomial setting. If you want to know how many trials it will take to observe a particular event, then we have a geometric setting.

3. Vocab Geometric Setting Geometric distribution Mean or expected value of a geometric random variable

4. Geometric Setting 1.“Success” or “Failure” 2.The probability of success is the same for each observation 3.All observations are independent 4.The variable of interest is the number of trials needed to obtain the first success

5. An experiment consists of rolling a single die. The random variable is defined as X = the number of trials until a 3 occurs. Geometric? 1.“3” or “not a three” 2.1/6 probability of rolling a 3 each time 3.Independent events 4. Variable of interest – number of rolls until a 3 occurs Yes, Geometric distribution

6. Rule for calculating Geometric Probabilities If X has a geometric distribution with p probability 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, then the probability that the first success occurs on the nth trial is

7. X1234567… P(X)p(1-p)p(1-p) 2 p(1-p) 3 p(1-p) 4 p(1-p) 5 p(1-p) 6 p… p is the probability that the success will occur on the 1 st observation. p(1-p) is the probability that the success will occur on the 2 nd observation. …

8. X12345 P(X).1667.1389.1157 Create the geometric distribution for the number of rolls of a die until a 3 occurs.

9. Calculator 2 nd vars D geometric pdf, enter, (.16667, L1) sto L2 L1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …) NO ZERO – it is not possible to find the probability of the first success occurring on the 0 observation. Look at the geometric pdf – this means that there is approx. a 9.6% chance that the first time a 3 is rolled will be the 4 th roll of the die.

10. Calculator What is the probability that it would take at most 6 rolls of the die to produce a 3. 2 nd vars E Geometric cdf, enter, (.1667,L1) sto in L3. There is approx a 67% chance that it would take at most 6 rolls of the die to roll a 3.

11. Mean The “expected value” or mean of the number of rolls of a die to roll a 3 is

12. The probability that it takes more than n trials to see the first success is: The probability that it will take more than 4 rolls of a die to roll a 3 is approx. 48%.