Geometric Distribution

Geometric distribution is a probability model that help us determine how many failures occur before a single success.

Random Variable

X = number of failures before success

Formula

q = probability of failure for a single trial ( = 1-p )
p = probability of success for a single trial
x = number of failures ( = 0, 1, 2, ...)
p+q = 1

Expectation of a Geometric Distribution

q = probability of failure for a single trial
p = probability of success for a single trial

Examples

1. Jason is rolling a die. Calculate the probability of getting a 1 on the 5th roll.



First. find "p" and "q"
p = probability of success for a single trial
p = getting a 1
p = 1/6

q = probability of failure for a single trial ( = 1-p )
q = getting anything but 1
q = 1 - p
q = 1 - 1/6
q = 5/6

Next, x is the number of failures before a success
In the question it said getting a 1 on the 5th roll, meaning the 5th roll is the success. Than there are 4 failures before it.
x = 4


The probability of getting a 1 on the 5th roll is 625 / 7776.


2. What is the expected number of rolls before a 1 come out?

p = probability of success for a single trial
p = getting a 1
p = 1/6

q = probability of failure for a single trial
q = getting anything but 1
q = 1 - 1/6
q = 5/6


The expected number of rolls before a 1 come out is 5


3. A top NHL hockey player scores on 93% of his shots in a shooting competition.

a) What is the probability that the player will not miss the goal until his 20th try?

First, find q.
- Since the questiong is asking, the player does not miss the goal UNTIL his 20th
try, this means he will miss on the 20th try. Therefore, the question is asking the
probability of the NHL player missing on his 20th shot. So, the player's
percentage of scoring is actually probability of failure.



Therefore,
q = 0.93


Second, find p
- Since we are given the probability of failure we must determine the probability of success by subtracting 1 - q.

p = probabilty of success in each single trial
p = 1-q
p = 1-0.93
p = 0.07

Therefore,
p = 0.07


Third, determine X
- Because the player misses the 20th shot, it means he scored 19 of the previous shots. Since the scoring of the player is measured as unsuccessful, the number of unsuccessful trials is 19.

Therefore,
x = 19


Forth, solve.

The probability of the NHL hockey player not missing until his 20th try is 0.0716%.


b) What is the expected number of shots before he misses?

First, find p & q
p = probability of success in each single
p = 0.07

q = probability of failure
q = 0.93

Next, Solve

Therefore, the NHL hockey player will shoot 13.3 shots before he misses.


By: Jason K., Shahnaz S., Andrew K.,
Improved By: Amrit L., Brendan S.


References:
Mr. D'Onofrio's notes, 7.3 handout.
Ms. Richardson's notes, 7.3 handout
Gr. 12 Data Textbook, pg. 395