Transcript theoretical probability

10-3 Theoretical Probability
Learn to estimate probability using
theoretical methods.
10-3 Theoretical Probability
Vocabulary
equally likely
theoretical probability
fair
geometric probability
mutually exclusive
disjoint events
10-3 Theoretical Probability
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.
10-3 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
10-3 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
10-3 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
10-3 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
10-3 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.
1H
2H
3H
4H
5H
6H
1T
2T
3T
4T
5T
6T
10-3 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
10-3 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
10-3 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
10-3 Theoretical Probability
Theoretical probability that is based on the
ratios of geometric lengths, areas, or
volumes is called geometric probability.
10-3 Theoretical Probability
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.
area of circle – area of triangle
probability =
area of circle
The probability that a point chosen within the
circle is within the shaded region is
10-3 Theoretical Probability
Check It Out: Example 3
Find the probability that a point
chosen randomly inside the circle is
within triangle. Round to the nearest
hundredth.
area of triangle
probability =
area of circle
The probability that a point chosen within the
circle is within the shaded region is
10-3 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.
10-3 Theoretical Probability
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.
10-3 Theoretical Probability
Additional Example 4 Continued
The event “total = 5” consists of 4 outcomes, (1, 4),
(2, 3), (3, 2), and (4, 1). So P(total = 5) = 4 .
36
The event “total = 2” consists of 1 outcome, (1, 1), so
P(total = 2) = 1 .
36
P(game ends) = P(total = 5) + P(total = 2)
= 4 + 1
36 36
The probability that the game will end is 5 , or about
36
13.9%.
10-3 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.
10-3 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%.
10-3 Theoretical Probability
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
10-3 Theoretical Probability
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
10-3 Theoretical Probability
Lesson Quiz for Student Response Systems
1. An experiment consists of spinning this spinner
once. Identify P(odd number).
A. 2
5
B. 2
C. 3
5
D. 3
10-3 Theoretical Probability
Lesson Quiz for Student Response Systems
2. An experiment consists of spinning this spinner
once. Identify P(not 8).
A.
1
2
B. 1
3
C. 2
3
D.
7
8
10-3 Theoretical Probability
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).
1
A.
2
B. 1
4
C. 3
4
D. 1