Download 05_Probability SSS Handout

Probability
You will want to be familiar with the probability formulas that are provided on the exam.
 When determining the probability of A or B, remember to subtract the intersection of A and B
P ( A  B )  P ( A)  P ( B )  P ( A  B )


When determining the probability of A given B, divide the intersection by the probability of
P( A  B)
B P( A | B) 
P( B)
The mean of a random variable is the sum of each value multiplied by the probability of each
outcome. E ( X )   x   xi pi

The variance of a random variable is the sum each deviation squared multiplied by the
probability of each outcome
Var ( X )   x2   ( xi   x ) 2 pi

Remember standard deviation is the square root of variance!

Remember for Binomial random variables, there should be two outcomes for each trial, each
trial should be independent of the rest, there should be a fixed number of trials, and the
probability of “success” should remain the same throughout.
o If X has a binomial distribution with number of trials, n, and probability of success, p,
then:
n
P ( X  k )    p k (1  p ) n  k
k 
 X  np
 X  np (1  p )
Additional rules (not included in test booklet):
 If two events are disjoint (mutually exclusive), then
P( A  B)  0
Example: Let A represent drawing a black card and B represent drawing a heart. Since hearts
are red,
P (Black card  Heart)  P ( A  B )
0

Two events are independent if and only if
P ( A | B )  P ( A) or P ( A  B )  P ( A) P ( B )
Example: Let A represent drawing a black card and B represent rolling a 5. Since cards in a standard
deck and six-sided dice are independent,
P (Black card  Rolling a Five)  P ( A  B )
 P ( A) P ( B )

11
 
26

1
12
Other useful tools:
Tree diagrams provide easy‐to‐interpret models for situations in which a scenario may be decomposed
into multiple stages, one following another. It is important to understand that the probability on each
branch in a tree diagram is a conditional probability.
Two‐way (contingency) tables are particularly useful in interpreting conditional probabilities and
examining independence. In such tables the entry in each cell is the joint probability for the row and
column events that define the cell. The total probability for all cells must be 1, and the sum for each row
(column) gives the “marginal” probability for that row (column). If you have a single row or column add
to 1 you may need to rethink the formulation of the table!
Example:
Taking AP
Physics
Not taking AP
Physics
Total
Taking AP Stats
Not taking AP
Stats
Total
0.45
0.35
0.80
0.15
0.05
0.20
0.60
0.40
1.00
One person’s name is drawn at random from this population. Determine each
1. The probability the person is taking AP Stats.
P(AP Stat.)  0.6
2. The probability the person is taking AP Stats or taking AP Physics.
P (AP Stat.  AP Phys.)  0.6  0.8  0.45  0.95
3. The probability the person is taking AP Stats and taking AP Physics.
P (AP Stat.  AP Phys.)  0.45
4. The person is taking AP Physics given that he/she is taking AP Stats.
0.45
P (AP Phys.|AP Stat.) 
 0.75
0.60
5. Are taking AP Stats and taking AP Physics independent? Explain.
?
P (AP Phys.)  P (AP Phys.|AP Stat.)
No, since 0.80  0.75 , taking AP Stats and taking AP Physics are not independent events.
Venn Diagrams are sometimes helpful, especially when given the probability of separate events.
Rules for Random Variables:
For any random variable X, where a and b are constants:
E ( aX  b)  a  E ( X )  b
Var(aX + b) = a2Var(X)
Example:
The expected value of a six-sided die is 3.5 and its variance is 2.91667. Suppose we want to multiply
the outcome of rolling a six-sided die by 5 and then add 2. In other words let Y  5 X  2 . Then the
expected value of Y is
E (Y )  E (5 X  2)
 5  E( X )  2
 5(3.5)  2
 19.5
And the variance of Y is
Var (Y )  Var ( aX  b )
 a 2Var ( X )
 25(2.91667)
 72.91667
When combining random variables X and Y, the means follow the same operation as the combination, in
other words, if you are adding the means add, if you are subtracting the means subtract:
E(X ± Y ) = E(X) ± E(Y )
For independent random variables X and Y, the variances always add. Be careful if you are given
standard deviation!:
Var(X ± Y ) = Var(X) + Var(Y )
Example:
Imagine a population of ice cream containers with a mean content of 64 ounces and standard
deviation of 2 ounces. Suppose we want to take a single scoop from each container. The scoop has a
mean of 4 ounces with a standard deviation of 0.5 ounces. After all the containers have one scoop
out of them, the mean amount in each container should be 60 ounces
E ( X  Y )  E ( X )  E (Y )
 64  4
 60 ounces
Remember that standard deviation is the square root of variance so the standard deviation of the
containers after one scoop is taken out is
 X Y  Var ( X  Y )
 Var ( X )  Var (Y )
 2 2  0.5 2
 2.062 ounces
Example:
Suppose you roll a six-sided die twice and Y is the sum of those two rolls. Here, the random variable
written as Y  X 1  X 2 denotes the result of doing the same underlying action twice, noting the value
obtained on the first action is independent of the one obtained on the second action, and adding the
two values together. The mean of Y is then
E (Y )  E ( X 1 )  E ( X 2 )
 3.5  3.5
7
And the standard deviation of Y is
 X1  X 2  Var ( X 1 )  Var ( X 2 )
 (2.916667)  (2.916667)
 2.415
Compare the distribution of Y with the distribution of K the result of rolling one die and doubling its
value. K  2 X so the mean of K is
E (K )  2E ( X )
 2(3.5)
7
And the standard deviation of K is
 2 X  2 2 Var ( X )
 4(2.916667)
 3.416
Thus X 1  X 2  2 X even though X , X 1 , and X 2 have the same distribution. From the examples,
you can see that X 1  X 2 and 2 X have the same expected value but a different variance and
standard deviation!
Vocabulary Reminder
The expected value of a random variable is the same as the mean of that random variable. Do not
confuse “expected value” with “most likely outcome”.
Calculator Use
To save time on the exam, you will want to use your calculator for probability computations.
Specifically, you will want to know how to:
• Enter a probability distribution in lists (values in one list and probabilities in a second list) and
use 1‐Var Stats to compute the mean (expected value) and standard deviation.
• Use NormCdf to calculate areas associated with normal curves and use invNorm to calculate
variable values associated with a given percentile.
• Use binomPdf to compute the probability of a specific binomial outcome and binomCdf to
compute the probability of an interval of binomial outcomes.
Note: When you use your calculator for computations on Free Response questions, it will be very
important to provide proper communication and support for your work.
Multiple Choice Questions:
Questions 1 and 2 refer to the following situation:
The class of 1968 and 1998 held a joint reunion in 2008 at the local high school. Attendees were asked
to fill out a survey to determine what they did after graduation. The results are summarized in the table
below
College Job Military Other
1968
56
73
85
7
1998
173
62
37
20
1.
What is the probability that a randomly selected attendee graduated in 1998 and went into the
military?
A.
B.
C.
D.
E.
0.072
0.127
0.303
0.596
0.669
2.
What is the probability that a randomly selected 1968 graduate went to college after graduation?
A.
B.
C.
D.
E.
0.245
0.253
0.560
0.592
0.755
3.
A fair die is rolled 3 times. The first 2 rolls resulted in 2 fives. What is the probability of not rolling a
five on the next roll?
A. 1
5
B.
6
2
 3  1  5
C.    
1 6  6
2
1 5
D.  
6 6
E. 0
4.
Which of the following statements is true for two events, each with probability greater than 0?
A.
B.
C.
D.
E.
If the events are mutually exclusive, they must be independent.
If the events are independent, they must be mutually exclusive.
If the events are not mutually exclusive, they must be independent.
If the events are not independent, they must be mutually exclusive.
If the events are mutually exclusive, they cannot be independent.
5.
Let X represent a random variable whose distribution is normal, with a mean of 50 and a standard
deviation of 5. Which of the following is equivalent to P ( X  55) ?
A.
B.
C.
D.
E.
P ( X  55)
P ( X  55)
P ( X  45)
P (45  X  55)
1  P( X  45)
6.
Lynn is planning to fly from New York to Los Angeles and will take the Airtight Airlines flight that
leaves at 8 A.M. The website she used to make her reservation states that the probability that the
flight will arrive in Los Angeles on time is 0.70. Of the following, which is the most reasonable
explanation for how that probability could have been estimated?
A. By using an extended weather forecast for the date of her flight, which showed a 30% chance of
bad weather
B. By making assumptions about how airplanes work, and factoring all of those assumptions into an
equation to arrive at the probability
C. From the fact that, of all airline flights arriving in California, 70% arrive on time
D. From the fact that, of all airline flight in the United States, 70% arrive on time
E. From the fact that, on all previous days this particular flight had been scheduled, it had arrived
on time 70% of those days
7.
According to a recent study on climate change, 40% of adults say that global climate change is a
threat to the well-being of their country. Suppose a random sample of 25 adults is interviewed, what
is the probability that more than 14 say that global climate change is a threat?
A.
B.
C.
D.
E.
0.0132
0.0344
0.0778
0.9222
0.9656
8.
In a game, a spinner with five equal-sized spaces is labeled from A to E. If a player spins an A they
win 15 points. If any other letter is spun the player loses 4 points. What is the expected gain or loss
from playing the game 40 times?
A.
B.
C.
D.
E.
Gain of 360 points.
Gain of 55 points.
Gain of 8 points.
Loss of 1 point.
Loss of 8 points.
For questions 9 and 10 refer to the information below:
The number of sweatshirts a vendor sells daily has the following probability distribution.
Number of Sweatshirts X
P(X)
0
0.3
1
0.2
2
0.3
3
0.1
4
0.08
5
0.02
9.
If each sweatshirt sells for $25, what is the expected daily total dollar amount taken in by the vendor
from the sale of sweatshirts?
A.
B.
C.
D.
E.
$5.00
$7.60
$35.50
$38.00
$75.00
10.
Suppose the vendor constructs a weekly sales report for the 7-day dollar amount taken in by the
vendor, Y. What is the standard deviation for Y, assuming  X  $1.33 , each sweatshirt sells for $25,
and the sales on each day are independent?
A.
B.
C.
D.
E.
$15.26
$17.60
$76.29
$87.99
$232.80
11.
Students, teachers, and parents in the drama club have been asked to decide on the fall fundraiser.
They must choose from selling popcorn, suckers, or beef jerky. Their votes are given in the table
below.
Students
Teachers
Parents
Total
Popcorn
35
10
45
90
Suckers
35
0
55
90
Beef Jerky
80
10
0
90
Which of the following statement(s) is true?
I.
II.
III.
A.
B.
C.
D.
E.
The events teacher and suckers are independent.
The events beef jerky and parents are mutually exclusive.
The probability of student or popcorn is approximately 0.7593
I only
II only
III only
I and III only
II and III only
Total
150
20
100
270
Free Response Questions:
12. Two medicines are available as relief for motion sickness for commercial airline pilots.


Medicine A is known to effectively cure the condition 40 percent of the time. A single dose
of medicine A costs $10.
Medicine B is known to effectively cure the condition 80 percent of the time. A single dose
of medicine B costs $40.
The medicines work independently of one another. Both medicines can be safely administered while
in flight. The airline company intends to recommend one of the following two plans as the standard
treatment for their pilots.


Plan I: Treat with medicine A first. If it is not effective, then treat with medicine B.
Plan II: Treat with medicine B first. If it is not effective, then treat with medicine A.
(a) If a pilot uses plan I, what is the probability that the pilot will be cured?
If a pilot uses plan II, what is the probability that the pilot will be cured?
(b) Compute the expected cost per pilot when plan I is used for treatment. Compute the expected
cost per pilot when plan II is used for treatment.
(c) Based on the results in parts (a) and (b), which plan would you recommend to the airline?
Explain your recommendation.
13. After winning the national championship, a local sports team is selling t-shirts. Each customer may
purchase up to 4 adult size shirts and 5 children's size shirts. Let A be the number of adult shirts
purchased by a single customer. The probability distribution of the number of adult shirts purchased
by a single customer is given in the table below.
A
P( A)
0
0.1
1
0.15
2
0.55
3
0.05
4
0.15
(a) Compute the mean and standard deviation for A.
(b) Suppose the mean and the standard deviation for the number of child shirts purchased by a single
customer are 2 and 1.5, respectively. Assume that the numbers of adult shirts and child shirts
purchased are independent random variables. Compute the mean and the standard deviation of
the total number of adult and child shirts purchased by a single customer.
(c) Suppose each child shirt costs $12 and each adult shirt costs $24. Compute the mean and
standard deviation of the total amount spent per purchase.
14. Men’s shoe sizes are determined by their foot lengths. Suppose that men’s foot-lengths are approximately
normally distributed with mean 27 centimeters and standard deviation 1.6 centimeters. A retailer sells men’s
sandals in sizes S, M, L, XL, where the sandal sizes are defined in the table below.
Sandal Size
S
M
L
Foot Length
24  foot length  26
26  foot length  28
28  foot length  30
XL
30  foot length  32
(a) Because the retailer only stocks the sizes listed above, what proportion of customers will find that the
retailer does not carry any sandals in their sizes? Show your work.
(b) Using a sketch of a normal curve, illustrate the proportion of men whose sandal size is L. Calculate this
proportion.
(c) Of 8 randomly selected customers, what is the probability that exactly 3 will request size L ? Show your
work.