Transcript 10-4 Theoretical Probability

10-4 Theoretical Probability
Warm Up
Problem of the Day
Lesson Presentation
Course 3
10-4 Theoretical Probability
Warm Up
1. If you roll a number cube, what are the
possible outcomes?
1, 2, 3, 4, 5, or 6
2. Add 1 + 1. 1
12
6
4
3. Add 1 + 2 . 5
2
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36
9
10-4 Theoretical Probability
Problem of the Day
A spinner is divided into 4 different-colored
sections. It is designed so that the
probability of spinning red is twice the
probability of spinning green, the probability
of spinning blue is 3 times the probability of
spinning green, and the probability of
spinning yellow is 4 times the probability of
spinning green. What is the probability of
spinning yellow?
0.4
Course 3
10-4 Theoretical Probability
Learn to estimate probability using
theoretical methods.
Course 3
10-4 Theoretical
Insert Lesson
Title Here
Probability
Vocabulary
theoretical probability
equally likely
fair
mutually exclusive
disjoint events
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10-4 Theoretical Probability
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
x=1
5
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10-4 Theoretical Probability
A coin, die, or other object is called fair if all
outcomes are equally likely.
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10-4 Theoretical Probability
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.
1
P(4) = number of outcomes for 4 =
5
5
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10-4 Theoretical Probability
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.
P(even number) = number of possible even numbers
5
2
=5
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10-4 Theoretical Probability
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.
1
P(1) = number of outcomes for 1 =
5
5
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10-4 Theoretical Probability
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.
P(odd number) = number of possible odd numbers
5
3
=5
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10-4 Theoretical Probability
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.
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1H
2H
3H
4H
5H
6H
1T
2T
3T
4T
5T
6T
10-4 Theoretical Probability
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).
6
1
P(tails) =
=
12
2
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10-4 Theoretical Probability
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).
2
1
P(head and tail) = 4 = 2
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10-4 Theoretical Probability
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).
1
P(both tails) =
4
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10-4 Theoretical Probability
Additional Example 3: Calculating Theoretical
Probability
Stephany has 2 dimes and 3 nickels. How
many pennies should be added so that the
3
probability of drawing a nickel is 7 ?
Adding pennies to the bag will increase the number
of possible outcomes. Let x equal the number of
pennies.
3
= 3
5+x
7
3(5 + x) = 3(7)
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Set up a proportion.
Find the cross products.
10-4 Theoretical Probability
Additional Example 3 Continued
15 + 3x = 21
–15
– 15
3x = 6
3
3
Multiply.
Subtract 15 from both sides.
Divide both sides by 3.
x= 2
2 pennies should be added to the bag.
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10-4 Theoretical Probability
Check It Out: Example 3
Carl has 3 green buttons and 4 purple buttons.
How many white buttons should be added so
that the probability of drawing a purple button
2
is 9 ?
Adding buttons to the bag will increase the number
of possible outcomes. Let x equal the number of
white buttons.
4
= 2
7+x
9
2(7 + x) = 9(4)
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Set up a proportion.
Find the cross products.
10-4 Theoretical Probability
Check It Out: Example 3 Continued
14 + 2x = 36
–14
– 14
2x = 22
11
11
Multiply.
Subtract 14 from both sides.
Divide both sides by 11.
x= 2
2 white buttons should be added to the bag.
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10-4 Theoretical Probability
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.
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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10-4 Theoretical Probability
Additional Example 4: Find the Probability of
Mutually Exclusive Events
Suppose you are playing a game in which you roll
two fair number cubes. 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?
The event “total = 2” consists of 1 outcome, (1, 1), so
P(total = 2) = 1 .
36
P(game ends) = P(total = 2) = 1
36
1
The probability that you will lose is 36 , or about 3%.
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10-4 Theoretical Probability
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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10-4 Theoretical Probability
Check It Out: Example 4 Continued
The event “both heads” consists of 1 outcome, (H, H),
so P(both heads) = 1 . The event “both tails” consists of
4
1
1 outcome, (T, T), so P(both tails) =
.
4
P(game ends) = P(both tails) + P(both heads)
=1+ 1
4 4
1
=2
1
The probability that the game will end is 2 , or 50%.
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10-4 Theoretical
Insert Lesson
Probability
Title Here
Lesson Quiz
An experiment consists of rolling a fair
number cube. Find each probability.
1
1. P(rolling an odd number)
21
2. P(rolling a prime number)
2
An experiment consists of rolling two fair
number cubes. Find each probability.
1
3. P(rolling two 3’s) 36
4. P(total shown > 10) 1
12
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