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Unit 6
Random
Variables
The weight of US men aged 19-39 is
approximately normally distributed with
mean of 194.7 lbs, stdev of 8.3 lbs.
Likewise for women: mean of 131.4 lbs
with mean of 7.5 lbs.
6 men and 6 women get on an elevator
with a max weight stated to be 2000 lbs.
What is the probability that the elevator
will be over its maximum carrying weight?
How would your answer change if you
found out that this was an express elevator
exclusively serving the marriage license
department in a local government facility?
6.3C Geometric Random Variables
 Determine if geometric conditions
have been met
 Compute & interpret geometric
probabilities
6.3C
Geometric Random Variables
 Determine if the conditions for a
geometric random variable have been met
 Find probabilities involving geometric
random variables
 Find the mean of a geom rand variable
Conditions for a Geometric Setting:
1.
2.
3.
4.
Binary—the possible outcomes are only
success and failure
Independent—Trials must be independent
(the result of one trial has no effect on the
results of another)
Trials unknown—Key word is “until” the
first success occurs. So the number of trials
will NOT be fixed.
Success—on each trial the prob p of
success must be the same
We are going to be counting the
number of trials UNTIL we encounter
a success. Therefore the possible
values are:
X = 1, 2, 3, 4, ……
Example: Roll a die and count the
number of rolls until you see a four
(a) What is the probability that it takes
3 rolls of the die until I see a four?
(b) What is the probability that it takes
8 rolls until I see a four?
General Formula for Geometric
Probabilities
Fail, Fail, Fail, Fail, Fail,….Success!
 Do we really need a formula?
Example: Monopoly
One way to get out of jail is to roll
doubles. Suppose that you have to
stay in jail until you do so. The
probability of rolling doubles is 1/6.
Example: Monopoly
(a) Explain why this is a geometric setting.
(b) Define the geometric random variable
and state its distribution.
(c) Find the probability that it takes exactly
three rolls to get out of jail.
(d) Find the probability that it takes more
than three rolls to get out of jail.
Using TI Technology:
geometpdf(p, k) computes 𝑃(𝑋 = 𝑘)
geometcdf(p, k) computes 𝑃 𝑋 ≤ 𝑘
You may NOT use calculator speak on the AP
exam. You must declare and define your
parameters as well as confirm the
geometric conditions.
Mean of a
Geometric Random Variable
(probability of success = p)
1
𝜇𝑋 = 𝐸 𝑋 =
𝑝
Example: Monopoly, part 2
(a) On average, how many rolls should it
take to escape from jail?
(b) Is this formula on the formula sheet?
(c) What is the probability that it takes
longer to escape from jail?
(d) What does this probability tell you
about the shape of this distribution?
All Probability Distributions
Discrete
Continuous
“John Doe”
Binomial
Geometric