Finite Math

1. Finite Math Section 8.5 Probability Distributions

2. 2. A die is rolled. If x is the number of spots on the die, which of the following is a probability distribution for x, the number of spots rolled? • x = 1 2 3 4 5 6 p(x) = 1/6 1/6 1/6 1/6 1/6 1/6 • x = 1 2 3 4 5 6 p(x) = 1/2 1/2 1/2 1/2 1/2 1/2 • x = 1 2 3 4 5 6 p(x) = 1/3 1/3 1/3 1/3 1/3 1/3

3. 3. Two coins are tossed. If x is the number of heads, which of the following is a probability distribution for x, the number of heads? • x = 1 2 p(x) = 1/2 1/2 • x = 0 1 2 p(x) = 1/4 1/2 1/4 • x = 0 1 2 p(x) = 1/3 1/3 1/3

4. 4. Three coins are tossed. If x is the number of heads, which of the following is a probability distribution for x, the number of heads? • x = 1 2 3 p(x) = 1/3 1/3 1/3 • x = 0 1 2 3 p(x) = 1/4 1/4 1/4 1/4 • x = 0 1 2 3 p(x) = 1/8 3/8 3/8 1/8

5. 5. Make your own empirical probability distribution by rolling two dice 36 times and keeping track of how many 2’s, 3’s, 4’s, 5’s, etc. using tally marks. An example is: • x = 2 3 4 5 6 7 8 9 10 11 12 tally I II I IIII III IIII IIII III IIII III I p(x)=1/36 2/36 1/36 4/36 3/36 4/36 4/36 3/36 4/36 3/36 1/36

6. 6. Two dice are rolled. If x is the number of spots, which of the following is the first part of the classical probability distribution for x, the number of spots? • x = 2 3 4 5 6 7 8 9 … p(x)=1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 … • x = 2 3 4 5 6 7 8 9 … p(x)=1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 … • x = 2 3 4 5 6 7 8 9 … p(x)=1/36 2/36 3/36 4/36 5/36 6/36 7/36 8/36 …

7. 7. The distribution of the number of heads when tossing 2 coins is: x = 0 1 2 P(x) = ¼ ½ ¼ What is the expected value of the number of heads? • ¼ • ½ • ¾ • 1 • 1 ¼ • 1 ½

8. 8. The distribution of the number of girls in a 3 child family is: x = 0 1 2 3 P(x) = 1/8 3/8 3/8 1/8 What is the expected number of girls, E(x) ? • ¼ • ½ • ¾ • 1 • 1 ¼ • 1 ½

9. 9. Two balls are drawn from a bag with 4 white balls and2 black balls. If x is the number of black balls drawn, which of the following is the probability distribution for x, the number of blacks? Record the correct answer to use on the next problem. • x = 1 2 3 4 p(x)= 1/8 3/8 3/8 1/8 • x = 0 1 2 p(x)= 1/4 1/2 1/4 • x = 0 1 2 p(x)=6/15 8/15 1/15 • x = 0 1 2 3 4 p(x)= 1/16 4/16 6/16 4/16 1/16

10. 10. Two balls are drawn from a bag with 4 white balls and 2 black balls. If x is the number of black balls drawn, what is the expected number of black balls, E(x)(expected value)? • 2/3 • 16/15 • 3/5

11. 11. • Similarly, if half of births are girls, then you would expect all 3 child families to average 1 ½ girls: ½ (3) = 1 ½ (even though you cannot have 1 ½ girls in a 3 child family).

12. 12. A delegation of 3 is selected from a city council made up of 5 liberals and 4 conservatives. What is the expected number of liberals in the delegation? • 1 • 4/3 • 5/3 • 2

13. 13. In a club with 20 senior members and 10 junior members, what is the expected number of seniors on a 3-member committee? • 1 • 4/3 • 5/3 • 2

14. 14. • 1/15 • 3/15 = 1/5 • 4/15 • 5/15 = 1/3 • 6/15 = 2/5 • 7/15 • 8/15

15. 15. • 1/15 • 3/15 = 1/5 • 4/15 • 5/15 = 1/3 • 6/15 = 2/5 • 7/15 • 8/15

16. 16. • 1/15 • 3/15 = 1/5 • 4/15 • 5/15 = 1/3 • 6/15 = 2/5 • 7/15 • 8/15

17. 17. • $0.00 • $53.33 1/3 • $13.33 1/3 • $66.66 2/3

18. 18. An insurance company has written 100 policies worth $10,000 each, 500 policies worth $5000, and 1000 policies worth $1000 for people of age 20. If the probability that a 20 year old person will die is .001, how much can the company expect to pay out during the year the policies were written? • $1,000 • $2,500 • $4,500

19. 19. a) A raffle offers a 1st prize of $1000, two 2nd prizes of $300 each and 20 3rd prizes of $10 each. If 10,000 tickets are sold at $1.00 each, find the expected winnings (using a probability distribution) for a person buying one ticket. b) Is this a fair game? • A) - $.82 B) No • A) - $.72 B) No • A) - $.62 B) No • A) $.82 B) Yes • A) $.72 B) Yes • A) $.62 B) Yes

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24. A • B • C • D