AP Statistics
Probability Review
Chapter 15: Probability Rules
Addition Rule:


When two events A and B are disjoint, we can use the addition rule for disjoint events:
P(A  B) = P(A) + P(B)
If two events are not disjoint, use the General Addition Rule:
For any two events A and B, P(A  B) = P(A) + P(B) – P(A  B)
Conditional Probability:

To find the probability of the event B given the event A, we restrict our attention to the total outcomes
in A. We then find the fraction of those outcomes in B that also occurred.
Multiplication Rule:


When two events A and B are independent, we can use the multiplication rule for independent events:
P(A  B) = P(A) x P(B)
If the two events are not independent, use the General Multiplication Rule:
For any two events A and B,
P(A  B) = P(A)  P(B|A)
or
P(A  B) = P(B)  P(A|B)
Independence:

Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are
independent whenever P(A|B) = P(A).)
Tree Diagram:



Figure 15.5 is a nice example of a tree diagram and shows how we multiply the probabilities of the
branches together.
All the final outcomes are disjoint and must add up to one.
We can add the final probabilities to find probabilities of compound events.
Chapter 16: Random Variables

A random variable assumes a value based on the outcome of a random event.
o We use a capital letter, like X, to denote a random variable.
o A particular value of a random variable will be denoted with the corresponding lower case
letter, in this case x.
o Two types of random variables:
 Discrete random variables can take one of a countable number of distinct outcomes.
 Example: Number of credit hours, amount of people at an event
 Continuous random variables can take any numeric value within a range of values.
 Example: Cost of books this term, daily temperature

A probability model for a random variable consists of:
o
The collection of all possible values of a random variable (x), and the probabilities P(x) that the
values occur.
o
Of particular interest is the value we expect a random variable to take on, notated μ (for
population mean) or E(X) for expected value.
 2  Var  X     x     P  x 
2
  SD  X   Var  X 

RECALL: Adding or subtracting a constant from data shifts the mean but doesn’t change the variance
or standard deviation:
E(X ± c) = E(X) ± c
Var(X ± c) = Var(X)

In general, multiplying each value of a random variable by a constant multiplies the mean by that
constant and the variance by the square of the constant (since we multiply the standard deviation by
the constant):
E(aX) = aE(X)

Var(aX) = a2Var(X)
In general,
o The mean of the sum of two random variables is the sum of the means.
o The mean of the difference of two random variables is the difference of the means.
E(X ± Y) = E(X) ± E(Y)
o
If the random variables are independent, the variance of their sum or difference is always the
sum of the variances.
Var(X ± Y) = Var(X) + Var(Y)
Chapter 17: Probability Models

We have Bernoulli trials if:
o there are two possible outcomes (success and failure).
o the probability of success, p, is constant.
o the trials are independent.

A Geometric probability model tells us the probability for a random variable that counts the number
of Bernoulli trials until the first success.
o

Geometric probability model for Bernoulli trials: Geom(p)

p = probability of success

q = 1 – p = probability of failure

X = number of trials until the first success occurs

P(X = x) = qx-1p
Calculator Shortcuts
o 2nd  DISTR  geometpdf(
o Note the pdf for Probability Density Function
o Used to find any individual outcome
o Format: geometpdf(p,x)
o
o
o
o

2nd  DISTR  geometcdf(
Note the cdf for Cumulative Density Function
Used to find the first success on or before the xth trial
Format: geometcdf(p,x)
A Binomial model tells us the probability for a random variable that counts the number of successes in a
fixed number of Bernoulli trials.
n!
Ck 
n
o In n trials, there are
k ! n  k ! ways to have k successes.
o Binomial probability model for Bernoulli trials: Binom(n,p)
 n = number of trials
 p = probability of success
 q = 1 – p = probability of failure
 X = # of successes in n trials
 P(X = x) = nCx px qn–x
  np
o
  npq
Calculator Shortcuts
 2nd  DISTR  binompdf(
 Note the pdf for Probability Density Function
 Used to find any individual outcome
 Format: binompdf(n,p,x)




2nd  DISTR  binomcdf(
Note the cdf for Cumulative Density Function
Used for getting x or fewer successes among n trials
Format: binomcdf(n,p,x)
Practice:
1. Suppose that 40% of cars in your area are manufactured in the United States, 30% in Japan, 10%
in Germany, and 20% in other countries. If cars are selected at random, find the probability that:
a) A car is not U.S.-made.
b) It is made in Japan or Germany.
c) You see two in a row from Japan.
d) None of three cars came from Germany.
e) At least one of three cars is U.S.-made.
f) The first Japanese car is the fourth one you choose.
2. One card is drawn from a standard deck of cards. What is the probability it is an ace or red?
3. Two cards are drawn without replacement. What is the probability they are both aces? Extend to
the probability of getting 5 hearts in a row.
4. I draw one card and look at it. I tell you it is red. What is the probability it is a heart? And what is
the probability it is red, given that it is a heart? (Conditional probability)
5. Are “red card” and “spade” independent? Mutually exclusive?
6. Are “red card” and “ace” independent? Mutually exclusive?
7. Are “face card” and “king” independent? Mutually exclusive?
8. A sample of automobiles dealerships found that 19% of automobiles sold are silver, 22% of
automobiles sold are SUVs. 16% of automobiles are silver SUVs.
a. Are the two events disjoint/mutually exclusive?
b. What is the probability that a randomly chosen sold automobile is silver or an SUV?
9. The following table shows the number of males and females wearing jeans to class on a certain
day.
c. What is the probability that a male wears jeans?
d. What is the probability that someone wearing jeans is a male?
e. Are being male and wearing jeans disjoint?
f. Are sex and attire independent?
10. Using the chart below:
Education majors Non-Education Majors
Total
Males
95
1015
1110
Females
700
1727
2427
Total
795
2742
3537
a. Find the probability the student is male or an education major.
b. Find the probability the student is female or not an education major.
c. Find the probability the student is not female or an education major.
d. Find the probability the student is a female and education major.
e. Given a student is male, what is the probability he is an education major?
11. April 2003, Science magazine reported on a new computer-based test for ovarian cancer, “clinical
proteomics,” that examines a blood sample for the presence of certain patterns of proteins.
Ovarian cancer, though dangerous, is very rare, afflicting only 1 of every 5000 women. The test is
highly sensitive, able to correctly detect the presence of ovarian cancer in 99.97% of women who
have the disease. However, it is unlikely to be used as a screening test in the general population
because the test gave false positives 5% of the time. Why are false positives such a big problem?
Draw a tree diagram and determine the probability that a woman who tests positive using this
method actually has ovarian cancer.
12. Consider a dice game: no points for rolling a 1, 2, or 3; 5 points for a 4 or 5; 50 points for a 6.
(b) Find the expected value and the standard deviation.
(c) Imagine doubling the points awarded. What are the new mean and standard deviation?
(d) Now imagine just playing the game twice. What are the mean and the standard deviation
of your total points?
(e) Suppose you and a friend both play the dice game. What are the mean and standard
deviation of the difference in your winnings?
13. Suppose a used car dealer runs autos through a two-stage process to get them ready to sell. The
mechanical checkup costs $50 per hour and takes an average of 90 minutes, with a standard
deviation of 15 minutes. The appearance prep (wash, polish, etc.) costs $6 per hour and takes an
average of 60 minutes, with a standard deviation of 5 minutes.
(f) What are the mean and standard deviation of the total time spent preparing a car? (Note
that we cannot find the standard deviation if we do not believe that the two phases of the
process are independent, an important assumption to check.)
(g) What are the mean and standard deviation of the total expense to prepare a car?
(h) What are the mean and standard deviation of the difference in costs for the two phases of
the operation?
(i) What is the probability that it will take longer to do the appearance prep than the
mechanical checkup? (Note that we cannot answer this question unless we believe each
phase of the process can be described by a Normal model.)
Practice Answers:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
(a) 0.6
(b) 0.4
(c) 0.09
7/13
1/221 ; 0.0005
½;1
No; Yes
Yes; No
No; No
(a) No – 16% are both
(b) 25%
(a) 12/17
(b) 3/5
(c) No
(a) 1810/3537
(b) 3442/3537
12. (a) µ = 10; 𝜎 2 = 325; 𝜎 = 18.03
(b) µ = 20; 𝜎 = 36.06
(c) µ = 20; 𝜎 = 25.5
(d) µ = 0; 𝜎 = 25.5
13. (a) µ = 150 min ; 𝜎 = 15.8 min
(b) µ = $81; 𝜎 = $12.51
(c) µ = $69; 𝜎 = $12.51
(d) µ = 30 min ; 𝜎 = 15.8 min
(d) 0.729
(d) No
(c) 1810/3537
(e) 0.784
(f) 0.1029
(d) 700/3537
(e) 95/1110