This document discusses theoretical probability and related concepts like experimental probability, odds, and complements. It provides examples of calculating theoretical probability for events like rolling dice or choosing marbles from a bag. It also shows how to convert between probabilities and odds. The objectives are to determine theoretical probability of an event and convert between probabilities and odds. Key terms defined include theoretical probability, equally likely, fair, complement, and odds.

1. UUNNIITT 99..66 TTHHEEOORREETTIICCAALL
PPRROOBBAABBIILLIITTYY

2. Warm Up
An experiment consists of spinning a
spinner 8 times. The spinner lands on
red 4 times, yellow 3 times, and green
once. Find the experimental probability
of each event.
1. The spinner lands on red.
2. The spinner does not land on green.
3. The spinner lands on yellow.

3. Objectives
Determine the theoretical probability
of an event.
Convert between probabilities and
odds.

4. Vocabulary
equally likely
theoretical probability
fair
complement
odds

5. When the outcomes in the sample space of an
experiment have the same chance of occurring,
the outcomes are said to be equally likely.

6. The theoretical probability of an event is the
ratio of the number of ways the event can occur to
the total number of equally likely outcomes.

7. An experiment in which all outcomes are equally
likely is said to be fair. You can usually assume
that experiments involving coins and number
cubes are fair.

8. Example 1A: Finding Theoretical Probability
An experiment consists of rolling a number
cube. Find the theoretical probability of each
outcome.
rolling a 5
There is one 5
on a number
cube.

9. Example 1B: Finding Theoretical Probability
An experiment consists of rolling a number
cube. Find the theoretical probability of each
outcome.
rolling an odd number
There 3 odd
numbers on
a cube.
= 0.5 = 50%

10. Example 1C: Finding Theoretical Probability
An experiment consists of rolling a number
cube. Find the theoretical probability of each
outcome.
rolling a number less than 3
There are 2
numbers
less three.

11. Check It Out! Example 1
An experiment consists of rolling a number cube.
Find the theoretical probability of each outcome.
a. Rolling an even number
There 3 even
numbers on
a cube.
b. Rolling a multiple of 3
= 0.5 = 50%
There are 2
multiples
of three.

12. Reading Math
The probability of an event can be written as
P(event). P(heads) means “the probability that
heads will be the outcome.”

13. When you toss a coin, there are two possible
outcomes, heads or tails. The table below shows
the theoretical probabilities and experimental
results of tossing a coin 10 times.

14. The sum of the probability of heads and the
probability of tails is 1, or 100%. This is because it is
certain that one of the two outcomes will always
occur.
P(event happening) + P(event not happening) = 1

15. The complement of an event is all the outcomes
in the sample space that are not included in the
event. The sum of the probabilities of an event and
its complement is 1, or 100%, because the event
will either happen or not happen.
P(event) + P(complement of event) = 1

16. Example 2: Finding Probability by Using the
Complement
A box contains only red, black, and white blocks.
The probability of choosing a red block is , the
probability of choosing a black block is . What is
the probability of choosing a white block?
P(red) + P(black) + P(white) = 100%
25% + 50% + P(white) = 100%
75% + P(white) = 100%
–75% –75%
P(white) = 25%
Either it will be a
white block or
not.
Subtract 75%
from both
sides.

17. Check It Out! Example 2
A jar has green, blue, purple, and white
marbles. The probability of choosing a green
marble is 0.2, the probability of choosing blue
is 0.3, the probability of choosing purple is 0.1.
What is the probability of choosing white?
Either it will be a white marble or not.
P(green) + P(blue) + P(purple) + P(white) = 1.0
0.2 + 0.3 + 0.1 + P(white) = 1.0
0.6 + P(white) = 1.0
– 0.6 – 0.6 Subtract 0.6 from
P(white) = 0.4
both sides.

18. Odds are another way to express the likelihood
of an event. The odds in favor of an event
describe the likelihood that the event will occur.
The odds against an event describe the likelihood
that the event will not occur.
Odds are usually written with a colon in the form
a:b, but can also be written as a to b or .

20. The two numbers given as the odds will add up to
the total number of possible outcomes. You can use
this relationship to convert between odds and
probabilities.

21. Reading Math
You may see an outcome called “favorable.” This
does not mean that the outcome is good or bad.
A favorable outcome is the outcome you are
looking for in a probability experiment.

22. Example 3A: Converting Between Odds and
Probabilities
The probability of rolling a 2 on a number
cube is . What are the odds of rolling a 2 ?
The probability of rolling a 2 is . There are 5 unfavorable
outcomes and 1 favorable outcome, thus the odds are 1:5.
Odds in favor are 1:5.

23. Example 3B: Converting Between Odds and
Probabilities
The odds in favor of winning a contest are 1:9.
What is the probability of winning the
contest?
The odds in favor of winning are 1:9, so the odds against are
9:1. This means there is 1 favorable outcome and 9
unfavorable outcomes for a total of 10 possible outcomes.
The probability of winning the contest is

24. Example 3C: Converting Between Odds and
Probabilities
The odds against a spinner landing on red are
2:3. What is the probability of the spinner
landing on red?
The odds against landing on red are 2:3, so the odds in
favor are 3:2. This means there are 3 favorable outcomes
and 2 unfavorable outcomes for a total of 5 possible
outcomes.
The probability of landing on red is

25. Check It Out! Example 3
The odds in favor of winning a free drink are
1:24. What is the probability of winning a
free drink?
The odds in favor of winning are 1:24, so the odds
against are 24:1. This means there is 1 favorable
outcome and 24 unfavorable outcomes for a total of 25
possible outcomes.
The probability of winning the free drink is

26. Lesson Quiz: Part I
Find the theoretical probability of each outcome.
1. Randomly choosing B from the letters in ALGEBRA
2. Rolling a factor of 10 on a number cube
3. The probability that it will be sunny is 15%. What
is the probability that it will not be sunny?
85%
4. The probability of choosing a red marble out of a
bag of marbles is . What are the odds in favor
of choosing a red marble? 3:7

27. Lesson Quiz: Part II
Find the theoretical probability of the outcome.
5. The odds against a spinner landing on blue are
7:5. Five sections of the spinner are red. What is
the probability of the spinner landing on red?

28. All rights belong to their
respective owners.
Copyright Disclaimer Under
Section 107 of the
Copyright Act 1976,
allowance is made for "fair
use" for purposes such as
criticism, comment, news
reporting, TEACHING,
scholarship, and research.
Fair use is a use permitted
by copyright statute that
might otherwise be
infringing.
Non-profit, EDUCATIONAL
or personal use tips the
balance in favor of fair use.