Theoretical Probability 10-6 Theoretical Probability Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Warm Up An experiment consists of spinning a spinner 8 times. The spinner lands on red 4 times, yellow 3 times, and green once. Find the experimental probability of each event. 1. The spinner lands on red. 2. The spinner does not land on green. 3. The spinner lands on yellow.

Objectives Determine the theoretical probability of an event. Convert between probabilities and odds.

Vocabulary equally likely theoretical probability fair complement odds

When the outcomes in the sample space of an experiment have the same 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.

An experiment in which all outcomes are equally likely is said to be fair. You can usually assume that experiments involving coins and number cubes are fair.

Example 1A: Finding Theoretical Probability An experiment consists of rolling a number cube. Find the theoretical probability of each outcome. rolling a 5 There is one 5 on a number cube.

Example 1B: Finding Theoretical Probability An experiment consists of rolling a number cube. Find the theoretical probability of each outcome. rolling an odd number There 3 odd numbers on a cube. = 0.5 = 50%

Example 1C: Finding Theoretical Probability An experiment consists of rolling a number cube. Find the theoretical probability of each outcome. rolling a number less than 3 There are 2 numbers less three.

Check It Out! Example 1 An experiment consists of rolling a number cube. Find the theoretical probability of each outcome. a. Rolling an even number There are 3 even numbers. = 0.5 = 50% b. Rolling a multiple of 3 There are 2 multiples of three.

The probability of an event can be written as P(event) The probability of an event can be written as P(event). P(heads) means “the probability that heads will be the outcome.” Reading Math

When you toss a coin, there are two possible outcomes, heads or tails When you toss a coin, there are two possible outcomes, heads or tails. The table below shows the theoretical probabilities and experimental results of tossing a coin 10 times.

The sum of the probability of heads and the probability of tails is 1, or 100%. This is because it is certain that one of the two outcomes will always occur. P(event happening) + P(event not happening) = 1

The complement of an event is all the outcomes in the sample space that are not included in the event. The sum of the probabilities of an event and its complement is 1, or 100%, because the event will either happen or not happen. P(event) + P(complement of event) = 1

Example 2: Finding Probability by Using the Complement A box contains only red, black, and white blocks. The probability of choosing a red block is , the probability of choosing a black block is . What is the probability of choosing a white block? P(red) + P(black) + P(white) = 100% Either it will be a white block or not. 25% + 50% + P(white) = 100% 75% + P(white) = 100% –75% –75% Subtract 75% from both sides. P(white) = 25%

Check It Out! Example 2 A jar has green, blue, purple, and white marbles. The probability of choosing a green marble is 0.2, the probability of choosing blue is 0.3, the probability of choosing purple is 0.1. What is the probability of choosing white? Either it will be a white marble or not. P(green) + P(blue) + P(purple) + P(white) = 1.0 0.2 + 0.3 + 0.1 + P(white) = 1.0 0.6 + P(white) = 1.0 – 0.6 – 0.6 Subtract 0.6 from both sides. P(white) = 0.4

Odds are another way to express the likelihood of an event Odds are another way to express the likelihood of an event. Odds express likelihood by comparing the number of ways an event can happen to the number of ways an event can fail to happen. The odds in favor of an event describe the likelihood that the event will occur. The odds against an event describe the likelihood that the event will not occur. Odds are usually written with a colon in the form a:b, but can also be written as a to b or .

The two numbers given as the odds will add up to the total number of possible outcomes. You can use this relationship to convert between odds and probabilities.

You may see an outcome called “favorable You may see an outcome called “favorable.” This does not mean that the outcome is good or bad. A favorable outcome is the outcome you are looking for in a probability experiment. Reading Math

Example 3A: Converting Between Odds and Probabilities The probability of rolling a 2 on a number cube is . What are the odds of rolling a 2 ? The probability of rolling a 2 is . There are 5 unfavorable outcomes and 1 favorable outcome, thus the odds are 1:5. Odds in favor are 1:5.

Example 3B: Converting Between Odds and Probabilities The odds in favor of winning a contest are 1:9. What is the probability of winning the contest? The odds in favor of winning are 1:9, so the odds against are 9:1. This means there is 1 favorable outcome and 9 unfavorable outcomes for a total of 10 possible outcomes. The probability of winning the contest is

Example 3C: Converting Between Odds and Probabilities The odds against a spinner landing on red are 2:3. What is the probability of the spinner landing on red? The odds against landing on red are 2:3, so the odds in favor are 3:2. This means there are 3 favorable outcomes and 2 unfavorable outcomes for a total of 5 possible outcomes. The probability of landing on red is

Check It Out! Example 3 The odds in favor of winning a free drink are 1:24. What is the probability of winning a free drink? The odds in favor of winning are 1:24, so the odds against are 24:1. This means there is 1 favorable outcome and 24 unfavorable outcomes for a total of 25 possible outcomes. The probability of winning the free drink is

Lesson Quiz: Part I Find the theoretical probability of each outcome. 1. Randomly choosing B from the letters in ALGEBRA 2. Rolling a factor of 10 on a number cube 3. The probability that it will be sunny is 15%. What is the probability that it will not be sunny? 85% 4. The probability of choosing a red marble out of a bag of marbles is . What are the odds in favor of choosing a red marble? 3:7

Lesson Quiz: Part II Find the theoretical probability of the outcome. 5. The odds against a spinner landing on blue are 7:5. Five sections of the spinner are red. What is the probability of the spinner landing on red?