Learn to estimate probability using theoretical methods'

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Learn to estimate probability using theoretical
methods.
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Vocabulary
theoretical probability equally
likely fair mutually exclusive
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Theoretical probability is used to estimate
probabilities by making certain assumptions about
an experiment. Suppose a sample space has 5
outcomes that are equally likely, that is, they
all have the same probability, x. The
probabilities must add to 1.
x x x x x 1
5x 1
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A coin, die, 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.
A. What is the probability of spinning a 4?
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Additional Example 1B Calculating Theoretical
Probability
An experiment consists of spinning this spinner
once.
B. What is the probability of spinning an even
number?
There are 2 outcomes in the event of spinning an
even number 2 and 4.
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Additional Example 1C Calculating Theoretical
Probability
An experiment consists of spinning this spinner
once.
C. What is the probability of spinning a number
less than 4?
There are 3 outcomes in the event of spinning a
number less than 4 1, 2, and 3.
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Try This Example 1A
An experiment consists of spinning this spinner
once.
A. What is the probability of spinning a 1?
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Try This Example 1B
An experiment consists of spinning this spinner
once.
B. What is the probability of spinning an odd
number?
There are 3 outcomes in the event of spinning an
odd number 1, 3, and 5.
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Try This Example 1C
An experiment consists of spinning this spinner
once.
C. What is the probability of spinning a number
less than 3?
There are 2 outcomes in the event of spinning a
number less than 3 1 and 2.
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Additional Example 2A Calculating Theoretical
Probability for a Fair Die and a Fair Coin
An experiment consists of rolling one fair die
and flipping a coin.
A. 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 Die and a Fair Coin
An experiment consists of rolling one fair die
and flipping a coin.
B. What is the probability of getting 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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Additional Example 2C Calculating Theoretical
Probability for a Fair Die and a Fair Coin
An experiment consists of rolling one fair die
and flipping a coin.
C. What is the probability of getting an even
number and heads?
There are 3 outcomes in the event getting an
even number and heads (2, H), (4, H), (6, H).
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Additional Example 2D Calculating Theoretical
Probability for a Fair Die and a Fair Coin
D. What is the probability of getting a prime
number?
There are 6 outcomes in the event getting a
prime number (2, T), (3, T), (5, T), (2, H),
(3, H), (5, H).
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Try This Example 2A
An experiment consists of flipping two coins.
A. Show a sample space that has all outcomes
equally likely.
The outcome of flipping two heads can be written
as HH. There are 4 possible outcomes in the
sample space.
HH TH HT TT
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Try This Example 2 Continued
An experiment consists of flipping two coins.
B. What is the probability of getting one head
and 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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Try This Example 2C
An experiment consists of flipping two coins.
C. What is the probability of getting heads on
both coins?
There is 1 outcome in the event both heads
(H, H).
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Try This Example 2D
An experiment consists of flipping two coins.
D. What is the probability of getting both tails?
There is 1 outcome in the event both tails (T,
T).
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• Two events are mutually exclusive if they cannot
  both occur in the same trial of an experiment.
  Suppose both A and B are two mutually exclusive
  events.
• P(both A and B will occur) 0
• P(either A or B will occur) P(A) P(B)

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Additional Example 3 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 the game
will end on your next roll?
It is impossible to roll a total of 5 and a total
of 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 3 Continued
P(game ends) P(total 5) P(total 2)
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Try This Example 3
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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Try This Example 3 Continued
P(game ends) P(both tails) P(both heads)
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Lesson Quiz
An experiment consists of rolling a fair die.
Find each probability. 1. P(rolling an odd
number) 2. P(rolling a prime number) An
experiment consists of rolling two fair dice.
Find each probability. 3. P(rolling two 3s) 4.
P(total shown gt 10)