Download Discrete Random Variables

AP statistics Chapter 6 Notes
Random Variables
6.1A
Name ________________________
Per ___ Date __________________
A probability model describes the possible outcomes of a chance process and the likelihood that
those outcomes will occur.
Definition: A random variable takes _____________________ that describe the outcomes of some
chance process.
Definition: The probability distribution of a random variable gives its _______________ and their
_________________ of occurring.
Discrete Random Variables
6.1A
Definition: A discrete random variable X takes a __________set of possible values with ________
between.
Definition: The probability distribution of a discrete random variable X lists the _________ xi and
their __________________ pi
Value x1 x2 x3 . . .
Prob. p1 p2 p3 . . .
The probabilities pi must satisfy two requirements to be legitimate:
a.) Every probability pi is between __ and __.
b.) The ____ of the probabilities is ____.
To find the probability of any event, add the probabilities pi of the particular values xi that make up
the event
6.1A
Sample: In 2010, there were 1319 games played in the National Hockey League’s regular
season. Imagine selecting one of these games at random and then randomly selecting one of the
two teams that played in the game. Define the random variable X = number of goals scored by a
randomly selected team in a randomly selected game. The table below gives the probability
distribution of X:
Problem:
(a) Show that the probability distribution for X is legitimate.
(b) Make a histogram of the probability distribution.
Describe what you see.
(c) What is the probability that the number of goals scored by
a randomly selected team in a randomly selected game is at
least 6?
6.1A
The mean (expected value) of any discrete random variable is an
__________of the possible outcomes, with each outcome ___________ by its probability.
Definition: Suppose X is a discrete random variable whose probability distribution is
Value x1 x2 x3 . . .
Prob. p1 p2 p3 . . .
To find the mean (expected value) of X, multiply each possible _______ by ___________________,
then add all of the products.
E ( X )   X  x1 p1  x2 p2  ...   xi pi
Interpretation of the mean:
“if we perform many, many, many trials we would expect the average (x-variable) to be _____”
6.1A
Since we use mean as the measure of center for a discrete random variable, we’ll use standard
deviation as our measure of spread. The definition of the variance of a random variable is similar
to the definition of the variance for a set of quantitative data.
Definition: Suppose X is a discrete random variable whose probability distribution is
Value x1 x2 x3 . . .
Prob. p1 p2 p3 . . .
And that _____ is the mean of X.


The Variance of X is
Var ( X )   X2  ( x1   X )2 p1  ( x2   X )2 p2  ...   ( xi   X )2 pi
To get the standard deviation of a random variable, take the
_____________ of the variance
 X  Var ( X )  ( x1   X )2 p1  ( x2   X )2 p2  ... 
(x  
i
X
) 2 pi
Interpretation of the standard deviation:
6.1A
“On average, the (x-variable) will differ from the mean of ____ by about _____”
In 1952, Dr. Virginia Apgar suggested five criteria for measuring a baby’s health at birth. She
developed a 0-1-2 scale to rate a newborn on each of the five criteria. A baby’s Apgar score is the
sum of the ratings on each of the five scales. What Apgar scores are typical? To find out, researchers
recorded the Apgar scores of over 2 million newborn babies in a single year. Imagine selecting one of
these newborns at random. (That’s our chance process.) Define the random variable X = Apgar score
of a randomly selected baby one minute after birth. The table below gives the probability distribution
for X
Problem: Compute the mean and standard deviation of the random variable X and interpret it in
context.
E( X )   X  0(0.001)  1(0.006)  2(0.007)  ...  10(0.053)  8.128
The mean Apgar score of a randomly selected newborn is 8.128. This is the long-term
average Apgar score of many, many randomly chosen babies.
 X  Var ( X )  (0  8.128)2 (0.001)  (1  8.128) 2 (0.006)  ...  (10  8.128)2 (0.053)
 2.066  1.437
On average, a randomly selected baby’s Apgar score will differ from the mean of 8.128 by
about 1.4.
6.1A
How to calculate mean and standard deviation of a discrete random variable on the calculator:
 Enter x-values into list 1
 Enter probability values into list 2
 Calculate the 1-variable stats with list 1 in the list category
 Use list 2 as the frequency list
 Calculate
x  mean , S x  standard deviation
6.1A
In 2010, there were 1319 games played in the National Hockey League’s regular season. Imagine
selecting one of these games at random and then randomly selecting one of the two teams that
played in the game. Define the random variable X = number of goals scored by a randomly selected
team in a randomly selected game. The table below gives the probability distribution of X:
Sample: Compute the mean and standard deviation of the random variable X and interpret this value
in context.
6.1A
In an experiment on the behavior of young children, each subject is placed in an area with five
toys. Past experiments have shown that the probability distribution of the number X of toys played
with by a randomly selected subject is as follows:
X toys
0
1
2
3
4
5
Probability
0.03 0.16 0.30 0.23 0.17 0.11
Sample: Calculate the mean and standard deviation of the random variable X and interpret this value
in context.
Continuous Random Variable
6.1B
Discrete random variables commonly arise from situations that require you to _________ something.
Situations that involve _______________ something often result in a continuous random variable.
Definition: A continuous random variable X takes on any value in an ___________ of values.
The ___________________________ of X is described by a density curve.
 The probability of any event is the _________ under the density curve and above/below the
____________ that make up the event.

The probability model of a ________ random variable X assigns a probability between 0 and 1
to each possible value of X.

A ___________ random variable Y has _____________ many possible values.

All continuous probability models assign probability of ___ to each individual outcome.

Only an _____________ of values will have a numerical (other than 0) probability.
6.1B
Sample: The heights of young women closely follow the Normal distribution with mean μ = 64 inches
and standard deviation σ = 2.7 inches. This is a distribution for a large set of data. Now choose one
young woman at random. Call her height Y. If we repeat the random choice very many times, the
distribution of values of Y is the same Normal distribution that describes the heights of all young
women. Find the probability that the chosen women is between 68 and 70 inches tall.
6.1B
Sample: Weights of Three-Year-Old Females
The weights of three-year-old females closely follow a Normal distribution with a mean of μ= 30.7
pounds and a standard deviation of σ = 3.6 pounds. Randomly choose one three-year-old female
and call her weight X. Find the probability that the randomly selected three-year-old female weighs at
least 30 pounds.
Linear Transformations of Random Variables
6.2A
Previous information – Chapter 2
1.
Adding or subtracting a constant (a) to each observation
 Adds a to the individual values and mean
 Does not add a to spread or change the shape
2. Multiplying or dividing each observation by a constant (b):
 Multiplies or divides by b to individual values and the mean
 Multiplies or divides by b to the spread
 Does not change the shape
6.2A
Do the rules apply to random variables? Let’s see.
Sample: Pete’s Jeep Tours offers a popular half-day trip in a
tourist area. There must be at least 2 passengers for the trip to
run, and the vehicle will hold up to 6 passengers. Define X as
the number of passengers on a randomly selected day.
The mean is _______ and the
standard deviation is ________
Transformation:
Pete charges $150 per passenger. The random variable C
describes the amount Pete collects on a randomly
selected day.
The mean is ____________and the
standard deviation is ___________
Compare the shape, center, and spread of the two
probability distributions.
6.2A
The Rules for multiplying or dividing a constant for Random Variables:
Multiplying or dividing each value of a random variable by a number b:
 Multiplies or divides the ____________ and ____________ by b
 Multiplies or divides the _________ by |b|
 Does not change the __________
Note: Multiplying a random variable by a constant b multiplies the variance by b2
Since variance is standard deviation2.
6.2A
Sample: El Dorado Community College
El Dorado Community College considers a student to be full-time if he or she is taking between 12
and 18 units. The number of units X that a randomly selected El Dorado Community College full-time
student is taking in the fall semester has the following distribution.
Number of Units (X) 12
13
14
15
16
17
18
Probability
0.25 0.10 0.05 0.30 0.10 0.05 0.15
What is the mean and standard deviation of the distribution?
At El Dorado Community College, the tuition for full-time students is $50 per unit. That is, if T =
tuition charge for a randomly selected full-time student, T = 50X.
What is the mean and standard deviation of the transformed distribution?
What would the distribution look like?
Tuition Charge (T)
Probability
6.2A
Do the rules still apply?
Sample: Consider Pete’s Jeep Tours again. We defined C as
the amount of money Pete collects on a randomly selected day.
The mean is __________ and the
standard deviation is ____________
Transformation: It costs Pete $100 per trip to buy permits, gas, and a ferry pass. The random
variable V describes the profit Pete makes on a randomly
selected day.
The mean is ____________ and the
standard deviation is _______________
Compare the shape, center, and spread of the two
6.2A
The Rules for Adding/Subtracting a constant for Random Variables:
Adding the same number a (which could be negative) to each value of a random variable:
 Adds a to the ________ and _____________
 Does not change the ______________
 Does not change the ____________
6.2A
Sample: El Dorado Community College
At El Dorado Community College, the tuition for full-time students is $50 per unit.
Here is the probability distribution for T and a histogram of the probability distribution:
Tuition Charge (T) 600 650 700 750 800 850 900
Probability
0.25 0.10 0.05 0.30 0.10 0.05 0.15
What is the mean and standard deviation of the distribution?
In addition to tuition charges, each full-time student at El Dorado Community College is assessed
student fees of $100 per semester. If C = overall cost for a randomly selected full-time student, C =
100 + T.
Here is the probability distribution for C and a histogram of the probability distribution:
What is the mean and standard deviation of the distribution?
What would the distribution look like?
Overall Cost (C)
Probability
The Rules for Linear Transformations on Random Variables:
Putting it all together
6.2A
6.2A
Effect on a linear transformation on the mean and standard deviation
If Y = a + bX is a linear transformation of the random variable X, then
 The probability distribution of Y has the same ________ as the probability distribution of X
 The mean Y 

The standard deviation  Y 

The variance VarY 
Sample: One brand of bathtub comes with a dial to set the water temperature. When the “babysafe”
setting is selected and the tub is filled, the temperature X of the water follows a normal distribution
with a mean of 34°C and a standard deviation of 2°C.
a.) Define the random variable Y to be the water temperature in degrees Fahrenheit (recall
that F 
9
x  32 ) when the dial is set on “babysafe.” Find the mean and standard deviation
5
of Y.
b.) According to Babies R Us, the temperature of a baby’s bathwater should be between 90°F
and 100°F. Find the probability that the water temperature on a randomly selected day when
the “babysafe” setting is used meets the babies r us recommendation. Show your work.
6.2A
Sample: In a large introductory statistics class, the distribution of X = raw scores on a test was
approximately normally distributed with a mean of 17.2 and a standard deviation of 3.8. The
professor decides to scale the scores by multiplying the raw scores by 4 and adding 10.
(a) Define the variable Y to be the scaled score of a randomly selected student from this class. Find
the mean and standard deviation of Y.
(b) What is the probability that a randomly selected student has a scaled test score of at least 90?
Combining two random variables
6.2B
Combining Random Variables: Adding and Subtracting Random Variables
Let’s investigate the result of adding and subtracting random variables.
Let X = the number of passengers on a randomly selected trip with Pete’s Jeep Tours. Let Y = the
number of passengers on a randomly selected trip with Erin’s Adventures.
var =
var =
What do you think the mean of X + Y would be?
What do you think the standard deviation of X + Y would be?
What do you think the variance of X + Y would be?
Let’s check to see if we are correct. Define T = X + Y.
What are the mean and standard deviation of T?
T=X+Y 2+2
2+3
2+4
2+5
3+2
3+3
4
5
6
7
5
6
.15x.3
.15x.4
.15x.2
.15x.1
.25x.3
.25x.4
P
3+4
7
3+5
8
4+2
6
4+3
7
4+4
8
.35+.4
.14
.35x.2
.07
.045
.06
.03
.015
.075
0.1
.25x.2
.05
.25x.1
.025
.35x.3
.105
T=X+Y
4+5
9
5+2
7
5+3
8
5+4
9
5+5
10
6+2
8
6+3
9
6+4
10
6+5
11
P
.35x.1
.035
.20x.3
.06
.20x.4
.08
.20x.2
.04
.20x.1
.02
.05x.3
.015
.05x.4
.02
.05x.2
.01
.05x.1
.005
T=X+Y
P
Mean =
4
0.045
5
0.135
6
0.235
7
0.265
8
0.19
9
0.095
10
0.03
11
.005
Standard Deviation =
Variance =
Can we add the means to get the mean of the sum?
Can we add the standard deviations to get the standard deviation of the sum?
6.2B
The only way to determine the probability for any value of T is if X and Y are ________________
random variables.
Definition: If knowing whether any event involving X alone has occurred tells us ________ about the
occurrence of any event involving Y alone, and vice versa, then X and Y are independent random
variables.
 Probability models often ____________ independence when the random variables describe
outcomes that appear ____________to each other.
 You should always ask whether the assumption of independence seems ________________.
In our investigation, it is reasonable to assume X and Y are independent since the siblings operate
their tours in ______________ parts of the country.
6.2B
Rule: Mean of the sum of random variables
For any two random variables X and Y, if T = X + Y, then the expected value of T is
ET  T 
In general, the mean of the sum of several random variables is the __________________________.
6.2B
Rule: Variance of the sum of random variables
When we add two independent random variables, their _________________ add
Standard deviations ______________ add.
For any two independent random variables X and Y, T = X + Y, then the variance (Var) of T is
VarT 
and since variance is standard deviation squared
 T2 
so we can find standard deviation by
T 
6.2B
In general, the variance of the sum of several independent random variables is the ________ of their
variances.
Sample: El Dorado Community College considers a student to be full-time if he or she is taking
between 12 and 18 units. The number of units X that a randomly selected El Dorado Community
College full-time student is taking in the fall semester has the following distribution.
The mean of this distribution is μ= 14.65 units, the variance is σ2 = 4.23 units2 and the standard
deviation is σ = 2.06 units.
El Dorado Community College also has a campus downtown, specializing in just a few fields of
study. Full-time students at the downtown campus only take 3-unit classes. Let Y = number of units
taken in the fall semester by a randomly selected full-time student at the downtown campus. Here is
the probability distribution of Y:
The mean of this distribution is μ= 15 units, the variance is σ2 =
5.40 units2 and the standard deviation is σ = 2.3 units.
If you were to randomly select 1 full-time student from the main campus and 1 full-time student from
the downtown campus and add their number of units, the expected value of the sum would be
The standard deviation of the sum would be
Sample: Let B = the amount spent on books in the fall semester for a randomly selected full-time
student at El Dorado Community College. Suppose that μB = 153 and σB = 32.
Recall from earlier that C = overall cost for tuition and fees for a randomly selected full-time student at
El Dorado Community College and μC = 832.50 and σC = 103.
Problem: Find the mean and standard deviation of the cost of tuition, fees, and books (C + B) for a
randomly selected full-time student at El Dorado Community College.
6.2B
Rule: Mean of the difference of random variables
For any two random variables X and Y, if D = X – Y, then the expected value of D is
ED  D 
In general, the mean of the difference of several random variables is the _________ of their means
**the order of the subtraction is important!
6.2B
Rule: Variance of the difference of random variables
For any two independent random variables X and Y, if D = X – Y, then the variance of D is
VarD 
and since variance is standard deviation squared
 D2 
so we can find standard deviation by
D 
In general, the variance of the difference of two independent random variables is the _________ of
their variances.
6.2B
Sample: Problem:
The mean of the main campus credits (T) is μ= 14.65 units, the variance is σ2 = 4.23 units2 and the
standard deviation is σ = 2.06 units.
The mean of the downtown campus tuition (U) is μ= 15 units, the variance is σ2 = 5.40 units2 and the
standard deviation is σ = 2.3 units.
Suppose we randomly select one full-time student from each of the two campuses. What are the
mean and standard deviation of the difference in tuition charges, D = T – U? Interpret each of these
values.
6.2B
Combining Normal distributions


If a random variable is normally distributed, we can use its mean and standard deviation to
compute probabilities.
An important fact about Normal random variables is that sum or difference of two independent
normal random variables is also _______________________.
6.2B
Sample : Mr. Starnes likes between 8.5 and 9 grams of sugar in his hot tea. Suppose the amount of
sugar in a randomly selected packet follows a normal distribution with mean 2.17g and a standard
deviation of 0.08g. If Mr. Starnes selects four packets at random, what is the probability that his tea
will taste right?
6.2B
Sample: Suppose that the height M of male speed daters follows a Normal distribution with a mean
of 70 inches and a standard deviation of 3.5 inches and the height F of female speed daters follows a
Normal distribution with a mean of 65 inches and a standard deviation of 3 inches. What is the
probability that a randomly selected male speed dater is taller than the randomly selected female
speed dater with whom he is paired?
6.3A
When the same chance process is repeated several times, we are often interested in whether a
particular outcome does or doesn’t happen on ______________. In some cases, the particular
number of repeated trials is FIXED IN ADVANCE and we are interested in the number of times a
particular event (called a “success”) _________. If the trials in these cases are independent and
each success has an equal chance of occurring, we have a binomial setting.
Binomial Distributions
Definition: A binomial setting arises when we perform several independent trials of the same
chance process and record the number of times that a particular outcome occurs. The four
conditions for a binomial setting are:




6.3A
B
I
Binary? The possible outcomes of each trial can be classified as ________________.
Independent? Trials must be independent; that is, knowing the outcome of one trial
must _____ have any effect on the outcome of any other trial.
N
Number? The _______________ n of the chance process must be determined in
advance.
S
Success? On each trial, the ______________ p of success must be the same
Definition: The count X of successes in a binomial setting is a binomial random variable.
Definition: The probability distribution of X is a binomial distribution with parameters n and p
where

n is the _________________ of the chance process

p is the ___________ of success on any one trial.
The possible values of X are the whole numbers from __________
Note: When checking the binomial condition, be sure to check the BINS and make sure you’re being
asked to count the number of successes in a certain number of trials.
6.3A
Samples: For each of the following situations, determine whether the given random variable has a
binomial distribution. Justify your answer.
1. Shuffle a deck of cards. Turn over the top card. Let X = the number of aces you observe. Put the
card back in the deck, and shuffle again. Repeat this process 10 times.
2. Choose three students at random from your class. Let Y = the number of students who are over 6
feet tall.
3. Flip a coin. If it's heads, roll a six-sided die. If it's tails, roll an eight sided die. Let W = the number
of fives you roll. Repeat this process 5 times.
6.3A
For any setting in which we are interested in k successes and n trials:
P(X = k) = P(exactly k successes in n trials)
= (number of arrangements) (probability of success)
Definition: The number of ways of ______________ k successes among n observations is given by
the binomial
coefficient
n
n!
n Ck also known as a _____________________!
 
 k  k !(n  k )!
For k = 0, 1, 2, . . .n where n! = n  n  1 n  2  ...  3 2 1 and 0! = 1
***Also known as a combination
Which can be found on your calculator under MATH, PRB, nCr
6.3A
The binomial coefficient counts the amount of ways in which k successes can be arranged among n
trials. The binomial probability P(X = k) is this count ____________ by the probability of any one
specific arrangement of the k successes.
Definition: Binomial probability
If X has the binomial distribution with n trials and probability p of success on each trial, the
possible values of X are 0, 1, 2, . . . , n. If k is anyone of these values,
n
P( X  k )    ____________   n Ck  _____________
k 
6.3A
Sample: Each child of a particular pair of parents has probability 0.25 of having blood type O.
Suppose the parents have 5 children
(a) Find the probability that exactly 3 of the children have type O blood.
(b) Should the parents be surprised if more than 3 of their children have type O blood?
6.3A
Sample: When rolling two dice, the probability of rolling doubles is 1/6. Suppose that a game player
rolls the dice 4 times, hoping to roll doubles.
(a) Find the probability that the player gets doubles twice in four attempts.
(b) Should the player be surprised if he gets doubles more than twice in four attempts? Justify your
answer
6.3A
.
To introduce her class to binomial distributions, Mrs. Desai gives a 10-item, multiple-choice quiz. The
catch is, students must simply guess an answer (A through E) for each question. Mrs. Desai uses her
computer’s random number generator to produce the answer key, so that each possible answer has
an equal chance to be chosen. Patti is one of the students in this class. Let X = the number of Patti’s
correct guesses.
1. Show that X is a binomial random variable.
2. Find and interpret P(X = 3)
3. To get a passing score on the quiz, a student must guess correctly at least 6 times. Would you be
surprised if Patti earned a passing score? Compute an appropriate probability to support your
answer.
6.3B
Describing a Binomial Distribution with center and spread
We describe the probability distribution of a binomial
random variable just like any other distribution – looking at
____________________________.
Consider the probability distribution of X = number of
children with type O blood in a family with 5 children.
Notice, the mean µX = _______ can be found another way.
Since each child has a 0.25 chance of inheriting type O blood,
we’d expect one-fourth of the five children to have this blood type.
That is, µX = ___________________. This method can be used to find the mean of any binomial
random variable with parameters n and p.
Definition:
Mean and Standard Deviation of a binomial distribution
If a count X has the binomial distribution with number of trials n and probability of success p , then the
mean and standard deviation of X are
X 
X 
Note: These formulas work only for ________distributions. They can’t be used for other distributions.
6.3B
Sample: To introduce her class to binomial distributions, Mrs. Desai gives a 10-item,multiple-choice
quiz. The catch is, students must simply guess an answer (A through E) for each
question. Mrs. Desai uses her computer’s random number generator to produce the answer key, so
that each possible answer has an equal chance to be chosen. Patti is one of the students in this
class. Let X = the number of Patti’s correct guesses.
1. Find μX. Interpret this value in context.
2. Find σx. Interpret this value in context.
3. What's the probability that the number of Patty's guesses is more than 2 standard deviations
above the mean? Show your method.
6.3B
The binomial distributions are important in statistics when we want to make inferences about the
___________ p of ________________ in a population.
Consider: Suppose 10% of CDs have defective copy-protection schemes that can harm computers.
A music distributor inspects an SRS of 10 CDs from a shipment of 10,000. Let X = number of
defective CDs. What is P(X = 0)? Note, this is not quite a binomial setting. Why?
The actual probability is P(no defective) =
9, 000 8,999
8,991

 ... 
 0.3485
10, 000 9,999
9,991
Using the binomial distribution, P( X  0)   10 C0  (.10)0  0.90   0.3487
10
In practice, the binomial distribution gives a ____________________ as long as we don’t sample
more than ___________ of the population.
10% Rule - Sampling without replacement condition:
When taking an SRS of size n from a population of size N, we can use a binomial distribution to
model the count of successes in the sample as long as
As n gets larger, something interesting happens to the shape of a binomial distribution. The figures
below show histograms of binomial distributions for different values of n and p. What do you notice as
n gets larger?
6.3B
Normal approximations for Binomial Distributions:
Suppose X has the binomial distribution with n trials and success probability of p. When n is large the
distribution of X is approximately normal with mean and standard deviation
X 
X 
As a rule of thumb, we will use the normal approximation when n is ___________________________
__________________ . That is, the expected number of successes and failures are both at least 10.
****Very important second semester*****
6.3B
Sample: Sample surveys show that fewer people enjoy shopping than in the past. A survey asked a
nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes,
but shopping is often frustrating and time-consuming.” Suppose that exactly 60% of all adult US
residents would say “Agree” if asked the same question. Let X = the number in the sample who
agree. Estimate the probability that 1520 or more of the sample agree.
Verify that X is approximately a binomial random variable.
B: Success = agree, Failure = don’t agree
I: Because the population of U.S. adults is greater than___________, it is reasonable to assume
the sampling without replacement condition is met.
N: ________________trials of the chance process
S: The probability of selecting an adult who agrees is ________________
Check the conditions for using a normal approximation.
Since ___________________________________ and ______________________________ are
both at ___________________, we may use the normal approximation.
Calculate P(X≥1520) using a normal approximation
P(X≥1520)
6.3B
Sample: In a survey of 506 teenagers ages 14-18, subjects were asked a variety of questions about
personal finance. One question asked teens if they had a debit card. Suppose that exactly 10% of
teens ages 14-18 have debit cards. Let X = the number of teens in a random sample of size 506 that
have a debit card.
(a) Show that the distribution of X is approximately binomial.
(b) Check the conditions for using a Normal approximation in this setting.
(c) Use a Normal distribution to estimate the probability that 40 or fewer teens in the sample have
debit cards.
Geometric Distributions
6.3C
Geometric Settings
In a binomial setting, the number of trials n is fixed and the binomial random variable X counts the
number of successes. In other situations, the goal is to repeat a chance behavior ____________
_________________. These situations are called geometric settings.
Definition: A geometric setting arises when we perform independent trials of the same chance
process and record the number of trials until a ______________________________________.
The four conditions are:




6.3C
6.3C
B
I
Binary? The possible outcomes of each trial can be classified as _________________
T
S
Trials? The goal is to count the number of trials until the _________________ occurs
Independent? Trials must be independent; that is, knowing the result of one trial must
not have ____________ on the result of any other trial.
Success? On each trial, the probability p of success must __________________.
Geometric Random Variable
The number of trials Y that it takes to get a success in a geometric setting is a geometric
random variable.
The probability distribution of Y is a geometric distribution with parameter p, the probability of a
success on any trial. The possible values of Y are 1, 2, 3, . . .
**Note: Like binomial random variables, it is important to be able to distinguish situation in which the
geometric distribution does and doesn’t apply.
Sample: The birthday game
A student will be selected at random from the class and asked to guess the day of the week
on which one of the teacher's friends was born, if the student guesses correctly, then the
class will have only one homework problem. If the student guesses incorrectly, the teacher
will again select a student from the class at random. This student will try to guess what day of
the week a DIFFERENT friend was born on. If they get it correct, the class only has to do two
problems.
The random variable of interest in this game is Y = the number of guesses it takes to correctly
identify the birth day of one of your teacher’s friends. What is the probability the first student guesses
correctly? The second? Third? What is the probability the kth student guesses corrrectly?
Verify that Y is a geometric random variable.
B: Success = correct guess, Failure = incorrect guess
I: The result of one student’s guess has no effect on the result of any other guess.
T: We’re counting the number of guesses up to and including the first correct guess.
S: On each trial, the probability of a correct guess is 1/7.
Calculate P(Y = 1), P(Y = 2), P(Y = 3), and P(Y = k)
P (Y = 1) = (1/7) = 0.1429
P (Y = 2) = (6/7)(1/7) = 0.1224
P (Y = 3) = (6/7)(6/7)(1/7) = 0.1050
Notice the pattern?
6.3C
Geometric Probability
If Y has the geometric distribution with probability of p of success on each trial, the possible
values of Y are 1, 2, 3, . . . If k is any one of these values
P(Y  k ) 
Sample: In the board game Monopoly, one way to get out of jail is to roll doubles.
(a) Find the probability that it takes 3 turns to roll doubles.
(b) Find the probability that it takes more than 3 turns to roll doubles and interpret this value in
context.
6.3C
Mean (Expected Value) of a Geometric Random Variable
If Y is a geometric random variable with probability p of success on each trial, then its mean
(expected value) is
E (Y )  Y 
Sample: In the board game Monopoly, one way to get out of jail is to roll doubles. If this was the
only way to get out of jail, how many turns would it take, on average?
6.3C
Determine whether each random variable described below satisfies the conditions for a
binomial setting, a geometric setting, or neither. Support your conclusion in each case.
(a) Draw a card from a standard deck of 52 playing cards, observe the card, return the card
to the deck, and shuffle. Count the number of times you draw a card in this manner until
you observe a jack.
(b) Joey buys a Virginia lottery ticket every week. X is the number of times in a year that he
wins a prize.
6.3C
Suppose that 20% of a herd of cows is infected with a particular disease.
(a) What is the probability that the first diseased cow is the 3rd cow tested?
(b) What is the probability that 4 or more cows would need to be tested until a diseased cow
was found?
6.3C
Research suggests that about 24% of 12-year-olds in the United States can pick out the
state of Colorado on a map.
(a) What is the probability that you must sample exactly 5 twelve-year-olds to find the first
one who can pick out Colorado on a map?
(b) What is the probability that you must sample 5 or more twelve-year-olds to find the first
one who can pick out Colorado on a map?