CK-12 Probability and Statistics - Advanced

4.6. The Geometric Probability Distribution http://www.ck12.org

p=Probability of anSoutcome

x=The number of trials until the firstSis observed

The figure below plots a few probability distributions of the Geometric distributions. Note how the curve starts high
and drops off, with lowerpvalues producing a faster drop-off.

Example:

A court is conducting a jury selection. Letxbe the number of prospective jurors who will be examined until one is
admitted as a juror for a trial. Suppose thatxis a geometric random variable andp, the probability of juror being
admitted, is 50%.

1. Find the mean and the standard deviation.


2. Find the probability that more than two prospective jurors must be examined before one is admitted to the
   jury.

Solution:

1. The mean and the standard deviation are,

μ=

1

p

=

1

0. 5

= 2

σ^2 =

1 −p

p^2

=

1 − 0. 5

0. 52

= 2

Thus

σ=

√

2 = 1. 41

2. To find the probability that more than two prospective jurors will be examined before one is selected,

p(x> 2 ) =p( 3 )+p( 4 )+p( 5 )+...

Obviously, this is an infinitely large sum so it is best to use the complementary rule: