Theoretical Probability: Finding the Probability of an Event

1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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. 1 4 1 3 7

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 1 2 ? 3 8 1 3 ? ? Ohio; Colorado; Oregon

4. Learn to find the theoretical probability of an event.

5. Vocabulary theoretical probability equally likely fair

6. Theoretical probability is used to find the probability of an event when all the outcomes are equally likely. Equally likelyoutcomes 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.

7. 9 20 = 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?Write your answer as a fraction, as a decimal, and as a percent. number of ways the event can occur total number of equally likely outcomes P = number of clear marbles total number of marbles P(clear) = Write the ratio. Substitute. Write as a decimal and write as a percent. = 0.45 = 45% The theoretical probability of drawing a clear marble is , 0.45, or 45%. 9 20

8. 11 20 = Additional Example 1B: Finding Theoretical Probability Find the probability of drawing a blue marble from the bag. number of ways the event can occur total number of equally likely outcomes P = number of blue marbles total number of marbles P(blue) = Write the ratio. Substitute. Write as a decimal and write as a percent. = 0.55 = 55% The theoretical probability of drawing a clear marble is , 0.55, or 55%. 11 20

9. 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? Write your answer as a fraction, as a decimal, and as a percent. number of ways the event can occur total number of equally likely outcomes P = number of green marbles total number of marbles P(green) = Write the ratio. Substitute. Write as a decimal and write as a percent. 8 20 = 0.4 = 40% = The theoretical probability of drawing a green marble is , 0.4, or 40%. 8 20

10. number of ways the event can occur total number of equally likely outcomes P = 2 numbers more than 4 6 possible outcomes P(number more than 4)= 2 6 = Check It Out: Example 1B Find the probability of rolling a number more than 4 on a fair number cube. 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. 1 3 =  0.33 33% The theoretical probability of rolling a number more than 4 is 0.33, or 33%. 1 3 ,

11. number of boys on the team number of members on the team P(boy) = 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 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. 13 23 P(boy)= Substitute.

12. Remember! The sum of the probabilities of an event and its complement is 1.

13. 13 23 Substitute for P(boy). 13 23 Subtract from both sides Simplify. Additional Example 2B: 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 girl’s name. P(boy) + P(girl) = 1 13 23 + P(girl) = 1 13 23 13 23 - = - 10 23 P(girl) =

14. number of girls in the class number of students in the class P(girl) = Check It Out: Example 2A 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? Find the theoretical probability. 12 27 Substitute. =

15. 12 27 Substitute for P(girl). 12 27 Subtract from both sides Simplify. Check It Out: Example 2B What is the theoretical probability that a boy’s name will be drawn? P(girl) + P(boy) = 1 12 27 + P(boy) = 1 12 27 12 27 - = - 15 27 P(boy) =

16. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

17. 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. 2 11 , 0.18, 18% 4 11 , 0.36, 36% , 0.64, 64% 7 11

18. Lesson Quiz for Student Response Systems 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.

19. Lesson Quiz for Student Response Systems 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.