1. 10-4 Theoretical Probability These are the notes that came with the teacher guide for the textbook we are using as a resource. These notes may be DIFFERENT than the notes we take in class if your teacher is in class, but should give you an idea of the concepts if you are struggling to understand them.

2. 10-4 Theoretical Probability 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.

3. 10-4 Theoretical Probability 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

4. 10-4 Theoretical Probability 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 1212 ? 3838 ? 1313 ?

5. 10-4 Theoretical Probability 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 ?

6. 10-4 Theoretical Probability Learn to find the theoretical probability of an event.

7. 10-4 Theoretical Probability Vocabulary theoretical probability equally likely fair

8. 10-4 Theoretical Probability Theoretical probability is used to find the probability of an event when all the outcomes are equally likely. Equally likely outcomes have the same 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.

9. 10-4 Theoretical Probability 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? 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(clear) = number of clear marbles total number of marbles = 0.45 = 45% 9 20 Write the ratio. Substitute. Write as a decimal and write as a percent. = 9 20 The theoretical probability of drawing a clear marble is, 0.45, or 45%.

10. 10-4 Theoretical Probability Find the probability of drawing a blue marble from the same bag from Example 1a. Example 1b: Finding 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. 10-4 Theoretical Probability PRACTICE 1a & 1b Write each answer as a fraction, as a decimal, and as a percent. 1a. Jane has 20 marbles in a bag. Of these 8 are green. Find the probability of drawing a green marble from the bag? 1b. Find the probability of rolling a number more than 4 on a fair number cube.

12. 10-4 Theoretical Probability P = number of ways the event can occur total number of equally likely outcomes P(green) = number of green marbles total number of marbles = 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. PRACTICE 1a – WORKED OUT Jane has 20 marbles in a bag. Of these 8 are green. Find the probability of drawing a green marble from the bag? Write your answer as a fraction, as a decimal, and as a percent. = 8 20

13. 10-4 Theoretical Probability 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, PRACTICE 1b – WORKED OUT 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.

14. 10-4 Theoretical Probability 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 random 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

15. 10-4 Theoretical Probability The sum of the probabilities of an event and its complement is 1. Remember!

16. 10-4 Theoretical Probability Example 2b: School Application 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.

17. 10-4 Theoretical Probability PRACTICE 2 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. 2a. If there are 15 boys and 12 girls in the class, what is the theoretical probability that a girl’s name will be drawn? 2b. What is the theoretical probability that a boy’s name will be drawn?

18. 10-4 Theoretical Probability PRACTICE 2a – WORKED OUT 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

19. 10-4 Theoretical Probability PRACTICE 2b – WORKED OUT What is the theoretical probability that a boy’s name will be drawn? + 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.

20. 10-4 Theoretical Probability Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

21. 10-4 Theoretical Probability 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.

22. 10-4 Theoretical Probability 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 4 11, 0.18, 18%, 0.36, 36%, 0.64, 64%

23. 10-4 Theoretical Probability 1. A number cube is rolled. Identify the probability of getting a number less than 4 as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. A. B. C. D. Lesson Quiz for Student Response Systems

24. 10-4 Theoretical Probability 1. A number cube is rolled. Identify the probability of getting a number less than 4 as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. A. B. C. D. Lesson Quiz for Student Response Systems

25. 10-4 Theoretical Probability 2. There are 40 balls in a bag. Of these 8 are white, 7 are blue, 12 are green, and 13 are yellow. Identify the probability of drawing a white ball as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. A. B. C. D. Lesson Quiz for Student Response Systems

26. 10-4 Theoretical Probability 2. There are 40 balls in a bag. Of these 8 are white, 7 are blue, 12 are green, and 13 are yellow. Identify the probability of drawing a white ball as a fraction, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. A. B. C. D. Lesson Quiz for Student Response Systems