1. Holt CA Course 1 8-3 Theoretical Probability SDAP3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1 – P is the probability of an event not occurring. California Standards

2. Holt CA Course 1 8-3 Theoretical Probability Vocabulary theoretical probability

3. Holt CA Course 1 8-3 Theoretical Probability In a board game, players use tiles with the letters of the alphabet to form words. Of the 125 tiles used in the game, 15 have the letter E on them. To determine the probability of drawing an E, you can draw tiles from a bag and record your results to find the experimental probability, or you can calculate the theoretical probability. Theoretical probability is used to find the probability of an event when all the outcomes are equally likely.

4. Holt CA Course 1 8-3 Theoretical Probability If each possible outcome of an experiment is equally likely, then the experiment is said to be fair. Experiments involving number cubes and coins are usually assumed to be fair.

5. Holt CA Course 1 8-3 Theoretical Probability Teacher Example 1: Finding Theoretical Probability Find the probability of rolling a number greater than 2 on a number cube (die). Write your answer as a ratio, a decimal, and a percent. P = number of ways the event can occur total number of equally likely outcomes P(>2) = number > 2 on a die total numbers on die = 0.666.. = 67% Write the ratio. Substitute. Write as a decimal and as a percent. = 4646 The theoretical probability of drawing a clear marble is, 0.666…, or 67%. 4646

6. Holt CA Course 1 8-3 Theoretical Probability P = number of ways the event can occur total number of equally likely outcomes P ( > 4) = 2 numbers greater than 4 6 possible outcomes = 2626 = 1313  0.33  33% The theoretical probability of rolling a number greater than 4 is 0.33, or 33%. 1313, Student Practice 1: Find the probability of rolling a number greater than 4 on a number cube. Write your answer as a ratio, a decimal, and a percent. Write the ratio. Substitute. Write as a decimal and as a percent.

7. Holt CA Course 1 8-3 Theoretical Probability Teacher Example 2: Finding Theoretical Probability Randomly choosing a red marble from a bag of 15 red, 15 blue, 15 green, 15 yellow, 15 black and 15 white marbles. Write your answer as a ratio, a decimal, and a percent. P = number of ways the event can occur total number of equally likely outcomes P(red) = number of red marbles total number of marbles = 0.1666.. = 17% Write the ratio. Substitute. Write as a decimal and as a percent. = 15 90 The theoretical probability of drawing a clear marble is, 0.166.., or 17%. 15 90

8. Holt CA Course 1 8-3 Theoretical Probability Student Practice 2: P = number of ways the event can occur total number of equally likely outcomes P(blue) = number of blue marbles total number of marbles = 11 20 = 0.55 = 55% The theoretical probability of drawing a blue marble is, 0.55, or 55%. 11 20 Write the ratio. Substitute. Write as a decimal and as a percent. Andy has 20 marbles in a bag. Of these, 9 are clear and 11 are blue. Find the probability of drawing a blue marble. Write your answer as a ratio, a decimal, and a percent.

9. Holt CA Course 1 8-3 Theoretical Probability Teacher Example 3A: School Application There are 13 boys and 10 girls on the track team. The name of each of the team members is written on an index card. A card is drawn at random to choose a student to run a sprint and the card is replaced in the stack. A. Find the theoretical probability of drawing a boy’s name. P(boy)= 13 23 P(boy) = number of boys on the team number of members on the team

10. Holt CA Course 1 8-3 Theoretical Probability The sum of the probabilities of an event and its complement is 1. Remember!

11. Holt CA Course 1 8-3 Theoretical Probability Teacher Example 3B: School Application B. Find the theoretical probability of drawing a girl’s name. + P(girl) = 1 13 23 P(boy) + P(girl) = 1 There are 13 boys and 10 girls on the track team. The name of each of the team members is written on an index card. A card is drawn at random to choose a student to run a sprint and the card is replaced in the stack. Substitute for P(boy). 13 23 Subtract from both sides. 13 23 – = – 13 23 13 23 P(girl) = 10 23 Simplify.

12. Holt CA Course 1 8-3 Theoretical Probability Student Practice 3A: = 12 27 P(girl) = number of girls in the class number of students in the class A. Find the theoretical probability that a girl’s name will be drawn. There are 15 boys and 12 girls in a class. A teacher has written the name of each student on a piece of paper and randomly draws a paper to determine which student will present the answer to the problem of the day.

13. Holt CA Course 1 8-3 Theoretical Probability Student Practice 3B: B. Find the theoretical probability that a boy’s name will be drawn. There are 15 boys and 12 girls in a class. A teacher has written the name of each student on a piece of paper and randomly draws a paper to determine which student will present the answer to the problem of the day. + P(boy) = 1 12 27 P(girl) + P(boy) = 1 Substitute for P(girl). 12 27 Subtract from both sides. 12 27 – = – 12 27 12 27 P(boy) = 15 27 Simplify.

14. Holt CA Course 1 8-3 Theoretical Probability Lesson Quiz Find the probabilities. Write your answer as a ratio, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. You have 11 cards, each with one of the letters from the word mathematics. 1. Find the probability of drawing an m from the pile of shuffled cards. 2. Find the probability of drawing a vowel. 3. Find the probability of drawing a consonant. 2 11, 0.18, 18% 4 11, 0.36, 36% 7 11, 0.64, 64%