## The Geometric Distribution

### The Geometric Setting:

1. Each observation falls into one of two categories: success or failure

2. The probability of a success, call it p, is the same for each observation.

3. Th observations are all independent.

4. The variable of interest is the number of trials required to obtain the first success.

Examles:

Flip a coin until you get a head

Roll a die until you get a 3

Attempt to get a three-point shot in basketball.

### Rule for Calculating Geometric Probabilites

P(X = n) =( (1- p)^n-1)p

### Example Using the TI-83

Construct a probability distribution table for X=number of rolls of a die until a 3 occurs:

X ___1____ 2______3_____4_____5____6_____7___. . .

P(X)_.1667_.1389_.1157__.0965__.0804_.0607_.0558__. . .

1. Enter the probability of success, 1/6 press Enter

2. Enter *(5/6) and press Enter

3. Continue to press Enter repeatedly.

### The mean of a geometric random variable

If X is a geometric random variable with probability of success p on each trial, then the mean, or expected value, of the random variable, that is, the expected number of trials required to get the first success, is m= 1/p.