Pre-Algebra 9-4 Theoretical Probability 9-4 Theoretical Probability Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation.

Pre-Algebra

9-4 Theoretical Probability9-4 Theoretical Probability

Pre-Algebra

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

9-4 Theoretical Probability

Warm Up

1. If you roll a number cube, what are the possible outcomes?

2. Add + .

3. Add + .

1, 2, 3, 4, 5, or 6

Pre-Algebra


336

16

12

236

1

4

5

9

Pre-Algebra


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

Pre-Algebra


Learn to estimate probability using theoretical methods.

Pre-Algebra


Vocabulary

theoretical probabilityequally likelyfairmutually exclusive

Pre-Algebra


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 = 15

Pre-Algebra


A coin, die, or other object is called fair if all outcomes are equally likely.

Pre-Algebra


An experiment consists of spinning this spinner once.

Additional Example 1A: Calculating Theoretical Probability

A. What is the probability of spinning a 4?

The spinner is fair, so all 5 outcomes

are equally likely. The probability of

spinning a 4 is P(4) = .15

Pre-Algebra



Additional Example 1B: Calculating Theoretical Probability

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.

P(spinning an even number) = number of possible even numbers52

5=

Pre-Algebra



Additional Example 1C: Calculating Theoretical Probability

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.

35P(spinning a number less than 4)=

Pre-Algebra



Try This: Example 1A

A. What is the probability of spinning a 1?

The spinner is fair, so all 5 outcomes

are equally likely. The probability of

spinning a 1 is P(1) = .15

Pre-Algebra



Try This: Example 1B

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.

P(spinning an odd number) = number of possible odd numbers53

5=

Pre-Algebra



Try This: Example 1C

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.

25P(spinning a number less than 3)=

Pre-Algebra


An experiment consists of rolling one fair die and flipping a coin.

Additional Example 2A: Calculating Theoretical Probability for a Fair Die and a Fair 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

Pre-Algebra


Additional Example 2B: Calculating Theoretical Probability for a Fair Die and a Fair 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).

P(tails) = = 612

12


Pre-Algebra


Additional Example 2C: Calculating Theoretical Probability for a Fair Die and a Fair 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).

P(even number and heads) = = 312

14


Pre-Algebra


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).

P(prime number) = = 612

12

Pre-Algebra


An experiment consists of flipping two coins.

Try This: Example 2A

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

Pre-Algebra


Try This: Example 2 Continued

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).

P(head and tail) = = 24

12


Pre-Algebra


Try This: Example 2C

C. What is the probability of getting heads on both coins?

There is 1 outcome in the event “both heads”: (H, H).

P(both heads) = 14


Pre-Algebra


Try This: Example 2D

D. What is the probability of getting both tails?

There is 1 outcome in the event “both tails”: (T, T).

P(both tails) = 14


Pre-Algebra


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)

Pre-Algebra


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.

Pre-Algebra


Additional Example 3 Continued

P(game ends) = P(total = 5) + P(total = 2)

The event “total = 5” consists of 4 outcomes, (1, 4), (2, 3), (3, 2), and (4, 1), so P(total = 5) = . The event “total = 2” consists of 1 outcome, (1, 1), so P(total = 2) = . 1

36

436

436

136= +

536

=

The probability that the game will end is , or about 13.9%.

536

Pre-Algebra


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.

Pre-Algebra


Try This: Example 3 Continued

P(game ends) = P(both tails) + P(both heads)

The event “both heads” consists of 1 outcome, (H, H),

so P(both heads) = . The event “both tails” consists of 1

outcome, (T, T), so P(both tails) = .

14 1

4

= +14

14

=12

The probability that the game will end is , or 50%.12

Pre-Algebra


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 3’s)

4. P(total shown > 10)

Pre-Algebra 9-4 Theoretical Probability 9-4 Theoretical Probability Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation.

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