Transcript theoretical probability

10-6 Theoretical Probability
Objectives
Determine the theoretical probability
of an event.
Convert between probabilities and
odds.
Holt McDougal Algebra 1
10-6 Theoretical Probability
When the outcomes in the sample space of an
experiment have the same chance of occurring,
the outcomes are said to be equally likely.
Holt McDougal Algebra 1
10-6 Theoretical Probability
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.
Holt McDougal Algebra 1
10-6 Theoretical Probability
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 (dice) are fair.
Holt McDougal Algebra 1
10-6 Theoretical Probability
Example 1A: Finding Theoretical Probability
An experiment consists of rolling a 6-sided die.
Find the theoretical probability of each
outcome.
rolling a 5
There is one 5
on a number
cube.
Holt McDougal Algebra 1
10-6 Theoretical Probability
Example 1B: Finding Theoretical Probability
An experiment consists of rolling a 6-sided die.
Find the theoretical probability of each
outcome.
rolling an odd number
There 3 odd
numbers on
a cube.
= 0.5 = 50%
Holt McDougal Algebra 1
10-6 Theoretical Probability
Example 1C: Finding Theoretical Probability
An experiment consists of rolling a 6-sided die.
Find the theoretical probability of each
outcome.
rolling a number less than 3
There are 2
numbers
less three.
Holt McDougal Algebra 1
10-6 Theoretical Probability
Check It Out! Example 1
An experiment consists of rolling a 6-sided die.
Find the theoretical probability of each outcome.
a. Rolling an even number
There are
3 even
numbers.
= 0.5 = 50%
b. Rolling a multiple of 3
There are 2
multiples
of three.
Holt McDougal Algebra 1
10-6 Theoretical Probability
Reading Math
The probability of an event can be written as
P(event). P(heads) means “the probability that
heads will be the outcome.”
Holt McDougal Algebra 1
10-6 Theoretical Probability
When you toss a coin, there are two possible
outcomes, heads or tails. 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
Holt McDougal Algebra 1
10-6 Theoretical Probability
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
Holt McDougal Algebra 1
10-6 Theoretical Probability
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% Either it will be a
white block or
25% + 50% + P(white) = 100%
not.
75% + P(white) = 100%
–75%
–75% Subtract 75%
from both
P(white) = 25%
sides.
Holt McDougal Algebra 1
10-6 Theoretical Probability
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
P(white) = 0.4
Holt McDougal Algebra 1
Subtract 0.6 from
both sides.
10-6 Theoretical Probability
Odds are another way to express the likelihood of
an event. Odds express likelihood by comparing
the number of ways an event can happen to the
number of ways an event can fail to happen. 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 .
Holt McDougal Algebra 1
10-6 Theoretical Probability
Holt McDougal Algebra 1
10-6 Theoretical Probability
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.
Holt McDougal Algebra 1
10-6 Theoretical Probability
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.
Holt McDougal Algebra 1
10-6 Theoretical Probability
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.
Holt McDougal Algebra 1
10-6 Theoretical Probability
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
Holt McDougal Algebra 1
10-6 Theoretical Probability
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
Holt McDougal Algebra 1
10-6 Theoretical Probability
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
Holt McDougal Algebra 1