PROBABILITY KEY
Example 1: Simple Probability
You roll a six-sided die whose sides are numbered from 1 through 6. Find:
(a) P(4) =
1
, good place to talk about changing fractions to decimals and percents, 0.16 , approx. 16.7%
6
(b) P(rolling an odd number) =
1
, 0.5, 50%
2
(c) P(rolling a number less than 5) =
(d) P(rolling a 1 or a 4) =
4 2
 = 0.6
6 3
2 1

6 3
(e) P(rolling a 1, 2, 3, 4, 5, or 6) = 1
(f) P(not 4) = 1 – P(4) =
5
6
(g) P(7) = 0
We learned a few of the very basic rules of probability in Example 1:
 When all outcomes are equally likely, the theoretical probability that an event A will occur is:
P(A) = number of outcomes in event A
total number of outcomes

The probability of an event occurring is between 0 and 1, inclusive.

The sum of the probabilities for all possible outcomes is equal to 1.

The probability that an event does not occur is 1 minus the probability that the event does occur:
P(A') = 1 – P(A).
Another way of saying this is that the probability of an event and its complement sum to 1: P(A) + P(A') = 1
Here are a few other basic probability rules that we will study in the examples that follow:

If A and B are two events, then P(A or B) = P(A) + P(B) – P(A and B).

If A and B are independent events, then P(A and B) = P(A) P(B).

If A and B are dependent events, then P(A and B) = P(A) P(B|A).
Discrete Random Variable
Def: A discrete random variable is the result of a count or statistical experiment that can take a countable, finite
number of values Examples: number of students in a class, bikes in the parking lot, outcomes of rolling a die,
number of sisters you have
Why are the following not discrete random variables? a person’s height, the time required to run a mile
Why is the concept of a discrete random variable important in the basic rules of probability?
1
Compound Events
Parts d) and e) from example 1 are examples of compound events. A compound event describes the probability
of two or more events occurring. There are two types of compound events: P(A or B) and the P(A and B).
P(A or B)
The outcomes corresponding to the probability of A or B can be represented by the union of A and B:
Shade the region that represents the union of A and B.
Sample Space
A
A and B
B
If A and B are two events, then the probability of A or B is:
P(A or B) = P(A) + P(B) – P(A and B)
Example 2: P(A or B) - Mutually Exclusive Events
A card is randomly selected from a standard deck of 52 cards. What is the P(ace or face card)? There are 4 aces
in a deck of cards and 12 face cards.
Show Venn Diagram and that events are mutually exclusive (do not overlap).
4 12 16 4


 = 0.308
52 52 52 13
Example 3: P(A or B) - NOT Mutually Exclusive Events
A card is randomly selected from a standard deck of 52 cards. What is the P(heart or face card)? There are 13
hearts in a deck of cards and 12 face cards. 3 cards are both a heart and a face card.
Show Venn Diagram and that events are NOT mutually exclusive (do overlap).
13 12 3 22 11




 0.423
52 52 52 52 26
Example 4: P(A or B) - NOT Mutually Exclusive Events
There are 300 seniors at Salt Lake City High. Forty of them take Calculus and fifty-five of them take Physics.
Of these, twenty-five students take both Calculus and Physics. What is the probability that a randomly selected
student takes Calculus or Physics?
Show Venn Diagram and that events are NOT mutually exclusive (do overlap).
40
55
25
70
7




 0.23
300 300 300 300 30
2
P(A and B)
The outcomes corresponding to the probability of A and B can be represented by the intersection of A and B:
Shade the region that represents the intersection of A and B.
Sample Space
A
A and B
B
Some compound events do not affect each other's outcomes, such as throwing a die and tossing a coin. These
are called independent events. If the outcome of one event affects the outcome of another, then the events are
called dependent events. For example, if you take two cards from a deck of playing cards, the likelihood of the
second card having a certain quality is altered by the fact that the first card has already been removed from the
deck.
If A and B are independent events, then the probability that both A and
B occur is:
P(A and B) = P(A) P(B)
If A and B are dependent events, then the probability that both A and B
occur is:
P(A and B) = P(A) P(B|A)
Example 5: P(A and B) - Independent Events
You are rolling a six-sided die and spinning the spinner shown below. Find P(even number and red).
red
blue
pink
white
3 1 3 1

  0.125
6 4 24 8
Example 6: P(A and B) - Independent Events
What is the probability that a coin will be flipped 4 times and the result will be HTHT?
1 1 1 1 1

 0.625
2 2 2 2 16
3
Example 7: Using a Probability Tree
A bag contains a total of 7 marbles. Five of the marbles are yellow and two of them are red. Draw a probability
tree to determine the different outcomes and the probability of each outcome occurring if you pull two marbles
out of the bag. Assume you replace the marble after the first pick.
5
Y
5 5 25
P(Y and Y) =

7
7 7 49
5
Y
2
5 2 2 5 20
R
7
Total Marbles
7
P(Y and R) =


5
Y
7 7 7 7 49
2
2 2 4
7
7
R
2
P(R and R) =

7 7 49
R
7
Point out that the branches of the tree always add to 1. In addition, point out that the sum of the outcomes = 1.
Example 8: P(A and B) - Dependent Events
A box contains seven green marbles, six blue marbles and eight orange marbles. Without looking, you choose
two marbles out of the bag. What is the probability that the first two picked will both be green if you don’t
replace the marble after the first pick?
7 6
42
1


 0.1
21 20 420 10
Example 9: P(A and B) - Dependent Events
A face-down deck of cards contains three hearts, six diamonds, four clubs and six spades. What is the
probability that the first two cards drawn will both be diamonds if the cards are not replaced after each pick?
6 5
30
5


 0.0877
19 18 342 57
Example 10: Using a Probability Tree
Copper Hills’ student population is 51% female. Of the females at the school, 58% eat school lunch. Of the
males at the school, 65% each school lunch. Make a probability tree and use it to find the probability that a
randomly selected student would eat school lunch.
65%
P(M and Y) = 49% 65% = 0.3185
Y
49%
M
35%
Total Population
51%
58%
N
Y
F
42%
P(F and Y) = 51% 58% = 0.2958
P(eat lunch) = 0.3185 + 0.2958 =
0.6143
N
Again, point out that branches sum to 1. You can show ALL outcomes and that they also sum to 1.
Example 11: Using Permutations and Combinations
What is the probability that 5 books placed randomly on a shelf will be placed alphabetically?
1
1

120
5 P5
4
Example 12: Using Permutations and Combinations
There are 25 kids in your class and your teacher is choosing 4 to help the secretary move books. If your teacher
randomly chooses 4 students, what is the probability that you and your 3 best friends will be selected (assuming
he/she is not biased)?
4 3 2 1
1
1
OR

25 24 23 22
12,650
25 C4
Example 13: Using Permutations and Combinations
Six boys and six girls belong to a club. Four officers are to be selected at random. What is the probability that
they will all be girls?
6 5 4 3
C4
15
1
OR


12 11 10 9
495 33
12 C4
6
Example 14: Using the Complement
Four high school friends will all be attending the same university next year. There are 14 dormitories on
campus. Find the probability that at least 2 of the friends will be in the same dormitory.
P(2 in the same dorm) + P(3 in the same dorm) + P(4 in the same dorm) OR
14 13 12 11
1 – P(none in the same dorm) = 1 –
=0.375
14 14 14 14
Basic Rules of Probability Summary

When all outcomes are equally likely, the theoretical probability that an event A will occur is:
P(A) = number of outcomes in event A
total number of outcomes

The probability of an event occurring is between 0 and 1, inclusive.

The sum of the probabilities for all possible outcomes is equal to 1.

The probability that an event does not occur is 1 minus the probability that the event does occur:
P(A') = 1 – P(A).
Another way of saying this is that the probability of an event and its complement sum to 1: P(A) + P(A') = 1

If A and B are two events, then P(A or B) = P(A) + P(B) – P(A and B).

If A and B are independent events, then P(A and B) = P(A) P(B).

If A and B are dependent events, then P(A and B) = P(A) P(B|A).
5
Additional Examples
1. A pet store contains 35 light green parakeets (14 females and 21 males) and 44 sky blue parakeets (28
females and 16 males). You randomly choose one of the parakeets. What is the probability that it is a
female or a sky blue parakeet?
42 44 28 58



 0.7341 Would also help to show in a 2-way table:
79 79 79 79
Female
Male
Light Green
14
21
Sky Blue
28
16
2. Of 162 students honored at an academic awards banquet, 48 won awards for mathematics and 78 won
awards for English. Fourteen of these students won awards for both mathematics and English. One of
the 162 students is chosen at random to be interviewed for a newspaper article. What is the probability
that the student interviewed won an award for English or Mathematics?
48 78 14 112



 0.6914
162 162 162 162
3. A weather forecaster says that the probability it will rain on Saturday or Sunday is 50%, the probability
it will rain on Saturday is 20%, and the probability it will rain on Sunday is 40%. What is the probability
that it will rain on both Saturday and Sunday?
0.1
Show Venn Diagram OR set up equation: 50% = 20% + 40% – x
4. The game show contestant spins a spinner with the letters G, I, Y and J on it, then either an easy or hard
question is picked randomly for her. What is the probability that the spinner will stop on the letter Y and
she is given an easy question?
1 1 1

4 2 8
5. Two cards are drawn from a standard 52-card playing deck. What is the probability that the draw will
yield an ace and a face card? Assume the first card is not replaced after you pick it. (Note: There are 4
aces and 12 face cards in a standard deck of cards.)
4 12
48
4


52 51 2652 221
6. You are about to attack a dragon in a role playing game. You will throw two dice, one numbered one to
eight and the other with the letters A through F. What is the probability that you will roll the values six
and A?
1 1 1

8 6 48
6
7. A bag contains a total of 8 marbles. There are only blue and green marbles in the bag. The following is a
probability tree showing the outcomes of pulling out a marble, replacing it, and then pulling out another
marble. G = green and B = blue. Complete the probability tree and use it to determine the probability of
pulling out a blue marble twice.
5 5 25
P(blue, blue) =

8 8 64
G
3
8
G
B
G
Total Marbles
B
B
8. A company is focus testing a new type of fruit drink. The focus group is 47% male. Of the males in the
group, 40% said they would buy the fruit drink, and of the females, 54% said they would buy the fruit
drink. Make a probability tree diagram and use it to find the probability that a randomly selected person
would buy the fruit drink.
40%
Y
P(M and Y) = 47% 40% = 0.188
47%
M
60%
Total Population
53%
54%
F
N
Y
P(F and Y) = 53% 54% = 0.2862
P(would buy) = 0.188 + 0.2862 =
0.4742
46%
N
9. The American Diabetes Association estimates that 5.9% of Americans have diabetes. Suppose that a
medical lab has developed a simple diagnostic test for diabetes that is 98% accurate for people who have
the disease and 95% accurate for people who do not have it. If the medical lab gives the test to a
randomly selected person, what is the probability that the diagnosis is correct?
0.952
10. At a particular company 64% of the employees are forty years old or over. Of those employees, 83% are
enrolled in the company’s retirement plan. Only 61% of the employees under forty years old are
enrolled in the plan. Make a probability tree diagram and use it to find the probability that a randomly
selected employee is enrolled in the company’s retirement plan.
0.751
11. You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The
player plays all 8 songs without repeating any song. What is the probability that the songs are played in
the same order they are listed on the CD?
1
1

= 0.0000248
40,320
8 P8
12. There are 16 teams in the state basketball tournament. If each team has an equal chance of winning,
what are the chances that the places will go as follows:
1st place – West Jordan HS
2nd place – Riverton
3rd place – Bingham
4th place – Copper Hills
1
1

43,680
16 P4
7
13. There are 204 students in the 10th grade. Five of these students will be selected randomly to represent
your class on a 5-person bowling team. What is the probability that the team chosen will be you and
your 4 best friends?
5 4 3 2 1
1
1
OR

204 203 202 201 200
2,802,350,040
204 C5
14. What are all the different ways the letters HTAM can be arranged? What is the probability that if you
randomly selected one of these arrangements, you would select the one that spells MATH?
1
1

24
4 P4
15. A local charity group is conducting a raffle where 50 tickets are to be sold – one per customer. There are
three prizes to be awarded. If the four organizers of the raffle each buy one ticket, what is the probability
that the organizers will win all of the prizes?
3 2 1 4 3 2
C3
4
OR

50 49 48 50 49 48
19,600
50 C3
16. Your English teacher is drawing names to see who will give their speech first. There are 26 students in
the class and 4 speeches will be given each day. What is the probability that you will give your speech
first? What is the probability that you will give your speech on the first day?
1
First =
26
4
C
2300
On the First Day =
or 3 25 
26
14950
26 C4
1
25 1
25 24 1
25 24 23 1
OR P(1st) + P(2nd) + P(3rd) + P(4th) =
(
)(
)(
)
26 26 25
26 25 24
26 25 24 23
17. A high school club has 10 members. The faculty advisor selects members at random to fill leadership
positions for president, vice president, treasurer, and secretary. Find the probability that Mark, one of the
club members, is selected for a leadership position.
0.4
18. Suppose that 5 people are chosen at random. Find the probability that at least two of the people share the
same birthday.
1 – P(none share the same birthday)
365 364 363 362 361
1–
= 1 – 0.9729 = 0.271 or 2.71%
365 365 365 365 365
4
19. The organizer of a potluck dinner sends 5 people a list of 8 different recipes and asks each person to
bring one of the items on the list. If all 5 people randomly choose a recipe from the list, what is the
probability that at least 2 will bring the same thing?
1 – P(all the dishes are different)
8 7 6 5 4
1–
= 1 – 0.2051 = 0.7949 or 79.49%
8 8 8 8 8
20. You collect movie trading cards, which have different scenes from a movie. For one movie there are 90
different cards in the set, and you have all of them except the final scene. To try and get this card, you
buy 10 packs of 8 cards each. All cards in a pack are different and each of the cards is equally likely to
be in a given pack. Find the probability that you will get the final scene.
1 – P(do not get the final scene)
C
1 – ( 89 8 )10 = 1 – 0.394197 = 0.606
90 C8
8
Conditional Probability
Some of the examples we just looked at are dependent events – the outcome of one event affects the outcome of
the other. To calculate the probability of these dependent events, we did the following:
If A and B are dependent events, then the probability that both A and B
occur is:
P(A and B) = P(A) P(B|A)
The conditional probability of an event B is the probability that event B will occur given the knowledge that
event A has already occurred. The formula for determining conditional probability can be derived from our
formula for calculating the probability of dependent events by simply rearranging the formula above:
P(B|A) = P(A and B)/ P(A)
Example 15: Conditional Probability
The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the
probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday:
P(student is absent | today is Friday)?
Use the Venn Diagram to show them whey we are dividing. When we did P(A and B), we multiplied the two
probabilities together to get the intersection of A and B. If we are given the intersection and the P(A), we can
see why we would be dividing to find the P(B given A).
A
A and B
B
P(student is absent | today is Friday) = P(today is Friday and a student is absent)/P(today is Friday)
0.03
 15%
0.2
Example 16: Conditional Probability
A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first
test. What percent of those who passed the first test also passed the second test?
0.25
 59.5%
0.42
Example 17: Conditional Probability – Data from a Table
Using the data in the table below, what is P(student’s favorite ice cream is Mint Chocolate Chip | student is a
senior)?
Year/Fav. Ice
Strawberry
Mint Choc. Chip
Oreo
Total
Cream
Cheesecake
52
78
106
236
Juniors
81
36
212
329
Seniors
133
114
318
565
Total
36
565  36  0.11
329 329
565
9
Additional Examples
1. At Kennedy Middle School, the probability that a student takes Technology and Spanish is 0.087. The
probability that a student takes Technology is 0.68. What is the probability that a student takes Spanish
given that the student is taking Technology?
13%
2. In New York State, 48% of all teenagers own a skateboard and 39% of all teenagers own a skateboard
and roller blades. What is the probability that a teenager owns roller blades given that the teenager owns
a skateboard?
81.25%
3. At a middle school, 18% of all students play football and basketball and 32% of all students play
football. What is the probability that a student plays basketball given that the student plays football?
56.25%
4. In the United States, 56% of all children get an allowance and 41% of all children get an allowance and
do household chores. What is the probability that a child does household chores given that the child gets
an allowance?
73.2%
5. Using the data in the table below, what is the probability that an endangered animal is a reptile?
Mammals
Birds
Reptiles
Amphibians
Other
59
75
14
9
198
Endangered
8
15
21
7
69
Threatened
67
90
35
16
267
Total
Total
355
120
475
14
 0.0394
355
10