Learn to estimate probability using theoretical methods.

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Learn to estimate probability using theoretical
methods.
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Vocabulary
equally likely theoretical probability fair geome
tric probability mutually exclusive disjoint
events
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When the outcomes in a sample space have an equal
chance of occurring, the outcomes are said to be
equally likely. 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.
A coin, number cube, or other object is called
fair if all outcomes are equally likely.
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Additional Example 1A Calculating Theoretical
Probability
An experiment consists of spinning this spinner
once. Find the probability of each event.
P(4)
The spinner is fair, so all 5 outcomes are
equally likely 1, 2, 3, 4, and 5.
number of outcomes for 4 5
P(4)
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Additional Example 1B Calculating Theoretical
Probability
An experiment consists of spinning this spinner
once. Find the probability of each event.
P(even number)
There are 2 outcomes in the event of spinning an
even number 2 and 4.
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Check It Out Example 1A
An experiment consists of spinning this spinner
once. Find the probability of each event.
P(1)
The spinner is fair, so all 5 outcomes are
equally likely 1, 2, 3, 4, and 5.
number of outcomes for 1 5
P(1)
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Check It Out Example 1B
An experiment consists of spinning this spinner
once. Find the probability of each event.
P(odd number)
There are 3 outcomes in the event of spinning an
odd number 1, 3, and 5.
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Additional Example 2A Calculating Probability
for a Fair Number Cube and a Fair Coin
An experiment consists of rolling one fair number
cube and flipping a coin. Find the probability of
the event.
Show a sample space that has all outcomes equally
likely.
The outcome of rolling a 5 and flipping heads can
be written as the ordered pair (5, H). There are
12 possible outcomes in the sample space.
1H 2H 3H 4H 5H 6H 1T 2T 3T 4T 5T 6T
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Additional Example 2B Calculating Theoretical
Probability for a Fair Coin
An experiment consists of flipping a coin. Find
the probability of the event.
P(tails)
There are 6 outcomes in the event flipping
tails (1, T), (2, T), (3, T), (4, T), (5, T),
and (6, T).
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Check It Out Example 2A
An experiment consists of flipping two coins.
Find the probability of each event.
P(one head one tail)
There are 2 outcomes in the event getting one
head and getting one tail (H, T) and (T, H).
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Check It Out Example 2B
An experiment consists of flipping two coins.
Find the probability of each event.
P(both tails)
There is 1 outcome in the event both tails (T,
T).
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Theoretical probability that is based on the
ratios of geometric lengths, areas, or volumes is
called geometric probability.
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Additional Example 3 Finding Geometric
Probability
Find the probability that a point chosen randomly
inside the circle is within the shaded region.
Round to the nearest hundredth.
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Check It Out Example 3
Find the probability that a point chosen randomly
inside the circle is within triangle. Round to
the nearest hundredth.
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Two events are mutually exclusive, or disjoint
events, if they cannot both occur in the same
trial of an experiment. For example, rolling a 5
and an even number on a number cube are mutually
exclusive events because they cannot both happen
at the same time.
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Additional Example 4 Find the Probability of
Mutually Exclusive Events
Suppose you are playing a game in which you roll
two fair dice. If you roll a total of five you
will win. If you roll a total of two, you will
lose. If you roll anything else, the game
continues. What is the probability that you will
lose on your next roll?
It is impossible to roll a total of 5 and a total
2 at the same time, so the events are mutually
exclusive. Add the probabilities to find the
probability of the game ending on your next roll.
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Additional Example 4 Continued
P(game ends) P(total 5) P(total 2)
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Check It Out Example 4
Suppose you are playing a game in which you flip
two coins. If you flip both heads you win and if
you flip both tails you lose. If you flip
anything else, the game continues. What is the
probability that the game will end on your next
flip?
It is impossible to flip both heads and tails at
the same time, so the events are mutually
exclusive. Add the probabilities to find the
probability of the game ending on your next flip.
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Check It Out Example 4 Continued
P(game ends) P(both tails) P(both heads)
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Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
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Lesson Quiz
An experiment consists of rolling a fair number
cube. Find each probability. 1. P(rolling an odd
number) 2. P(rolling a prime number) An
experiment consists of rolling two fair number
cubes. Find each probability. 3. P(rolling two
3s) 4. P(total shown gt 10)
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Lesson Quiz for Student Response Systems
1. An experiment consists of spinning this
spinner once. Identify P(odd number). A. B. 2
C. D. 3
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Lesson Quiz for Student Response Systems
2. An experiment consists of spinning this
spinner once. Identify P(not 8). A. B.
C. D.
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Lesson Quiz for Student Response Systems
3. An experiment consists of tossing two fair
coins at the same time. Identify P(at least one
head). A. B. C. D. 1