Download 10-4 - TeacherWeb

10-4 Theoretical Probability
Warm Up
Problem of the Day
Lesson Presentation
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
Course 3
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
Course 3
10-4 Theoretical Probability
A coin, die, or other object is called fair if all
outcomes are equally likely.
Course 3
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
Course 3
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
Course 3
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
Course 3
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
Course 3
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.
Course 3
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
Course 3
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)
Course 3
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.
Course 3
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)
Course 3
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%.
Course 3
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
Course 3