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The Geometric Distribution (Waiting Times)
Can we answer the following questions:
1. What is the probability that you can roll a sum of seven on a pair of dice in fewer than three
rolls?
2. If a traffic light flows at a given rate, what is the expected length of time that a pedestrian must
wait before crossing the street?
The probability model that enables us to answer these questions is called
A
distribution has a specified number of independent trials with two possible
outcomes,
or
. The random variable is the number of unsuccessful
outcomes before a success occurs.
Note that the number of trials prior to the first success is called the
before success.
period/time
If X is a random variable representing the waiting time, then the Geometric Distribution is given by
where p is the probability of success on each trial and q=1-p is the probability of failure on each
trial and x=0,1,2,...
The expected value of a random variable X that is geometrically distributed is
Example 1: In a repeated rolling of a pair of dice,
a) what is the probability that the first roll of doubles occurs on the third roll?
b) what is the expected waiting time (the number of rolls) before you roll doubles?
Solution: p=6/36=1/6 and q=1-p=5/6.
Example 2: Jamaal has a success rate of 68% for scoring on free throws in basketball. What is the
expected waiting time before he misses the basket on a free throw?
Solution:
Example 3: In a gambling game a player tosses a coin until a head is uppermost. He then receives
$2n, where n is the number of tosses.
a) what is the probability that the player receives $8.00 in one play of the game?
b) If the player must pay $5.00 to play, what is the win/loss per game?
Solution:
Example 4: What is the probability of rolling a sum of seven in fewer than three rolls of a pair of
dice?
Solution:
Example 5: Suppose that an intersection you pass on your way to school has a traffic light that is
green for 40 s and then amber or red for a total of 60 s.
a) What is the probability that the light will be green when you reach the intersection at least once
a week?
b) What is the expected number of days before the light is green when you reach the intersection?
Solution:
Homework: Worksheet.