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AP Statistics
AP Review Probability
Name __________________
Vocabulary
Law of large numbers
Simulation
Probability model
Complement
General addition rule
Union
Independent
General multiplication rule
Conditional probability
Probability
Sample space
Event
Mutually exclusive (disjoint)
Intersection
Conditional probability
Tree diagram
Multiplication rule for independent events
Summary
 A chance process has outcomes that we cannot predict but that nonetheless have a regular distribution in
very many repetitions. The law of large numbers says that the proportion of times that a particular
outcome occurs in many repetitions will approach a single number. This long-run relative frequency of a
chance outcome is its probability. A probability is a number between 0 (never occurs) and 1 (always
occurs).
 Probabilities describe only what happens in the long run. Short runs of random phenomena like tossing
coins or shooting a basketball often don’t look random to us because they do not show the regularity that
emerges only in very many repetitions.
 A simulation is an imitation of chance behavior, most often carried out with random numbers. To perform
a simulation, follow this four-step process:
1. Ask a question of interest about some chance process
2. Describe how to use a chance device to imitate one repetition of the process. Tell what you will record
at the end of each repetition.
3. Perform many repetitions of the simulation.
4. Use the results of your simulation to answer the question of interest.
 A probability model describes chance behavior by listing the possible outcomes in the sample space and
giving the probability that each outcome occurs.
 An event is a subset of the possible outcomes in the sample space. To find the probability that an event A
happens, we can rely on some basic probability rules.
 For any event A, 0 < P(A) < 1.
 P(S) = 1, where S = the sample space
 If all outcomes in the sample space are equally likely,
number of outcomes corresponding to event A
P  A 
total number of outcomes in sample space
 

Complement rule: P AC  1  P  A  where AC is the complement of event A; that is the event that A does

not happen.
Addition rule for mutually exclusive events: Events A and B are mutually exclusive (disjoint) if they have no
outcomes in common. If A and B are disjoint, P  A or B  P  A  P B.

A two-way table or a Venn diagram can be used to display the sample space for a chance process Two-way tables
and Venn diagrams can also be used to find probabilities involving events A and B, like the union  A  B  and
intersection  A  B . The event A  B (A or B) consists of all outcomes in event A, event B, or both. The event

A  B (A and B) consists of outcomes in both A and B.
The general addition rule can be used to find P  A or B : P  A or B  P  A  B  P  A  P  B  P  A  B

If one event has happened, the chance that another event will happen is a conditional probability. The notation

You can calculate conditional probabilities with the conditional probability formula P  A|B  

The general multiplication rule states that the probability of events A and B occurring together is

When chance behavior involves a sequence of outcomes, a tree diagram can be used to describe the sample space.
Tree diagrams can also help in finding the probability that two or more events occur together. We simply multiply
along the branches that correspond to the outcomes of interest.

When knowing that one event has occurred does not change the probability that another event happens,
we say that the two events are independent. For independent events A and B, P  A|B  P  A and
P  A|B represents the probability that event B occurs given that event A has occurred.
P  A  B
P B
P  A and B  P  A  B  P  A  P B| A
P  B| A  P B .


If two events A and B are mutually exclusive (disjoint), they cannot be independent.
In the special case of independent events, the multiplication rule becomes P  A and B  P  A  B  P  A  P B
Problems
1. The probability of drawing a jack, queen, or king from a standard deck of playing cards is approximately
0.23.
A. Explain what this probability means in the context of drawing from a deck of cards
If you were to repeatedly draw cards, with replacement, from a shuffled deck, you would draw a jack,
queen, or king 23% of the time.
B. Does this mean if we repeatedly draw a and, replace it, shuffle, and draw again 100 times that we will
draw a jack, queen, or king 23 times? Why or why not?
No, we’d expect to draw a jack, queen, or king 23 times out of 100, but we are not guaranteed exactly
23.
2. A popular airline knows that, in general, 95% of individuals who purchase a ticket for a 10-seat commuter
flight actually show up for the flight. In an effort to ensure a full flight, the airline sells 12 tickets for each
flight. Design and carry out a simulation to estimate the probability that the flight will be overbooked, that
is, more passengers show up than there are seats on the flight.
Let digits 01-95 represent a passenger showing up for the flight.
Let digits 96-00 represent a “no show”
Select 12 two digit numbers from a line of the random number table or a random number generator.
Ignore repeats until you have 12 numbers selected. Count how many times 96-00 occurs. If 96-00 occurs
less than two times, the flight is overbooked. Repeat this procedure 20 times. Determine how many of
the 20 trials result in an overbooked flight.
3. Consider drawing a card from a shuffled fair deck of 52 playing cards.
A. How many possible outcomes are in the sample space for this chance process? What’s the probability
for each outcome?
1
There are 52 possible outcomes. Each has a probability of
52
Define the following events: A: the card drawn is an Ace, B: the card drawn is a heart
4
13
P  A 
P B 
B. Find P(A) and P(B).
52
52
C. What is P(AC)?
P  AC   1  P  A   1 
4 48

52 52
D. Are the events A and B mutually exclusive? Why or why not?
No, A and B are not mutually exclusive because a card can be both an Ace and a heart.
E. Use a two-way table to display the sample space.
A
AC
1
12
B
BC
3
36
F. Use a Venn diagram to display the sample space.
G. Find P  A  B . Show your work.
P  A  B  P  A  P B  P  A  B 
4 13 1 16
 

52 52 52 52
4. Is there a relationship between gender and candy preference? Suppose 200 high school students were
asked to complete a survey about their favorite candies. The table below shows the gender of each
student and their favorite candy.
Male
Female
Total
Skittles
80
60
140
M&Ms
40
20
60
Total
120
80
200
Define A to be the event that a randomly selected student is male and B to be the event that a randomly
selected student likes Skittles. Are the events A and B independent? Justify your answer.
120
80
 0.60 P  A|B  
 0.5714
If A and B are independent, P  A   P  A|B  P  A  
200
140
Since these probabilities are not the same, A and B are not independent.
5. A study of high school juniors in three districts – Lakeville, Sheboygan, and Omaha – was conducted to
determine enrollment trends in AP mathematics courses – Calculus and Statistics. 42% of students in the
study came from Lakeville, 37% came from Sheboygan, and the rest came from Omaha. In Lakeville, 64%
of juniors took Statistics and the rest took Calculus. 58% of juniors in Sheboygan and 49% of juniors in
Omaha took Statistics while the rest took Calculus in each district. No juniors took both Statistics and
Calculus.
A. Describe this situation using a tree diagram and find the probability that a randomly selected student
from in the study took Statistics.
0.64
S = 0.42(0.64) = 0.2688
0.36
C
0.58
S = 0.37(0.58) = 0.2146
0.42
C
0.49
S = 0.21(0.49) = 0.1029
0.51
C
0.42
0.37
0.21
P  Statistics   0.2688  0.2146  0.1029  0.5863
B. Find P(student is from Lakeville | took Statistics).
P  student is from Lakeville | took Statistics  
0.2688
 0.4585
0.5863