1. 11-4 Theoretical Probability Course 2 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Warm Up 1. Two coins are tossed. What is the probability of getting two heads? 2. Give the probability that the roll of a number cube will show 1 or 4. 3. Give the expected number of rolls that will result in a 2 if a number cube is rolled 42 times. 1414 1313 7 Course 2 11-4 Theoretical Probability

3. Problem of the Day The name of a U.S. state is spelled out with letter tiles. Then the tiles are placed in a bag, and one is picked at random. What state was spelled out if the probability of picking the letter O is Ohio; Colorado; Oregon 1212 ? 3838 ? 1313 ? Course 2 11-4 Theoretical Probability

4. Learn to find the theoretical probability of an event. Course 2 11-4 Theoretical Probability

5. Vocabulary theoretical probability Insert Lesson Title Here Course 2 11-4 Theoretical Probability

6. In the game of Scrabble ®, players use tiles bearing the letters of the alphabet to form words. Of the 100 tiles used in a Scrabble game, 12 have the letter E on them. What is the probability of drawing an E from a bag of 100 Scrabble tiles? Course 2 11-4 Theoretical Probability

7. 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. Course 2 11-4 Theoretical Probability

8. 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. Course 2 11-4 Theoretical Probability

9. Find the probability. Write your answer as a fraction, as a decimal, and as a percent Additional Example 1A: Finding Theoretical Probability Andy has 20 marbles in a bag. Of these, 9 are clear and 11 are blue. Find the probability of drawing a clear marble from the bag? P = number of ways the event can occur total number of equally likely outcomes P(clear) = number of clear marbles total number of marbles = 9 20 = 0.45 = 45% The theoretical probability of drawing a clear marble is, 0.45, or 45%. 9 20 Write the ratio. Substitute. Write as a decimal and write as a percent. Course 2 11-4 Theoretical Probability

10. Find the probability of drawing a blue marble from the bag. Additional Example 1B: Finding Theoretical Probability Course 2 11-4 Theoretical Probability 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 clear marble is, 0.55, or 55%. 11 20 Write the ratio. Substitute. Write as a decimal and write as a percent.

11. Find the probability. Write your answer as a fraction, as a decimal, and as a percent. P = number of ways the event can occur total number of equally likely outcomes P(green) = number of green marbles total number of marbles = 8 20 = 0.4 = 40% The theoretical probability of drawing a green marble is, 0.4, or 40%. 8 20 Write the ratio. Substitute. Write as a decimal and write as a percent. Check It Out: Example 1A Jane has 20 marbles in a bag. Of these 8 are green. Find the probability of drawing a green marble from the bag? Course 2 11-4 Theoretical Probability

12. Find the probability of rolling a number more than 4 on a fair number cube. P = number of ways the event can occur total number of equally likely outcomes P (number more than 4) = 2 numbers more than 4 6 possible outcomes = 2626 = 1313  0.33  33% The theoretical probability of rolling a number more than 4 is 0.33, or 33%. 1313, Check It Out: Example 1B For a fair number cube, each of the six possible outcomes is equally likely. There are 2 ways to roll a number greater than 4: 5 or 6. Course 2 11-4 Theoretical Probability

13. Additional Example 2A: 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 randomly to choose a student to run a sprint and the card is replaced in the stack. Find the theoretical probability of drawing a boy’s name. P(boy)= 13 23 Substitute. P(boy) = number of boys on the team number of members on the team Course 2 11-4 Theoretical Probability

14. Course 2 11-4 Theoretical Probability The sum of the probabilities of an event and its complement is 1. Remember!

15. Additional Example 2B: School Application Find the theoretical probability of drawing a girl’s name. + P(girl) = 1 13 23 P(boy) + P(girl) = 1 Course 2 11-4 Theoretical Probability 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 randomly 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.

16. Check It Out: Example 2A Insert Lesson Title Here A teacher has written the name of each student on a piece of paper and placed the names in a box. She randomly draws a paper from the box to determine which student will present the answer to the problem of the day. If there are 15 boys and 12 girls in the class, what is the theoretical probability that a girl’s name will be drawn? = 12 27 Find the theoretical probability. Substitute. P(girl) = number of girls in the class number of students in the class Course 2 11-4 Theoretical Probability

17. Check It Out: Example 2B Insert Lesson Title Here = 15 27 Find the theoretical probability. Substitute. If there are 15 boys and 12 girls in the class, what is the theoretical probability that a boy’s name will be drawn? P(boy) = number of boys in the class number of students in the class Course 2 11-4 Theoretical Probability A teacher has written the name of each student on a piece of paper and placed the names in a box. She randomly draws a paper from the box to determine which student will present the answer to the problem of the day.

18. Lesson Quiz Find the probabilities. Write your answer as a fraction, 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. 7 11 2 11 Insert Lesson Title Here 4 11, 0.18, 18%, 0.36, 36%, 0.64, 64% Course 2 11-4 Theoretical Probability