Theoretical Probability

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10-6
Theoretical Probability
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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• 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.
  

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Objectives
Determine the theoretical probability of an
event. Convert between probabilities and odds.
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Vocabulary
equally likely theoretical probability
fair complement odds
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When the outcomes in the sample space of an
experiment have the same chance of occurring, the
outcomes are said to be equally likely.
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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.
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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.
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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.
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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
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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.
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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.
0.5 50
b. Rolling a multiple of 3
There are 2 multiples of three.
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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.
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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
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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
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Example 2 Finding Probability by Using the
Complement
P(red) P(black) P(white) 100
Either it will be a white block or not.
25 50 P(white) 100
75 P(white) 100
Subtract 75 from both sides.
P(white) 25
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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
Subtract 0.6 from both sides.
P(white) 0.4
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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.
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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.
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Example 3A Converting Between Odds and
Probabilities
Odds in favor are 15.
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Example 3B Converting Between Odds and
Probabilities
The odds in favor of winning a contest are 19.
What is the probability of winning the contest?
The odds in favor of winning are 19, so the odds
against are 91. This means there is 1 favorable
outcome and 9 unfavorable outcomes for a total
of 10 possible outcomes.
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Example 3C Converting Between Odds and
Probabilities
The odds against a spinner landing on red are
23. What is the probability of the spinner
landing on red?
The odds against landing on red are 23, so the
odds in favor are 32. This means there are 3
favorable outcomes and 2 unfavorable outcomes for
a total of 5 possible outcomes.
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Check It Out! Example 3
The odds in favor of winning a free drink are
124. What is the probability of winning a free
drink?
The odds in favor of winning are 124, so the
odds against are 241. This means there is 1
favorable outcome and 24 unfavorable outcomes for
a total of 25 possible outcomes.
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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?
37
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Lesson Quiz Part II
Find the theoretical probability of the outcome.
5. The odds against a spinner landing on blue are
75. Five sections of the spinner are red. What
is the probability of the spinner landing on red?