Question

read each question carefully and follow all instructions exactly. show all work; otherwise no credit. round to 4 decimals. 1. in a previous year, the weights of the members of the san francisco 49ers and the dallas cowboys were published in the san jose mercury news. the factual data were compiled into the following table. for the following, suppose that you randomly select one player from the 49ers or cowboys a. find the probability that his shirt number is from 1 to 33. b. find the probability that he weighs at most 210 pounds. c. find the probability that his shirt number is from 1 to 33 and he weighs at most 210 pounds. d. find the probability that his shirt number is from 1 to 33 or he weighs at most 210 pounds. e. out of the members that are at most 210 pounds, find the probability that their shirt number is from 1 - 33 2. suppose that you have eight cards. five are green and three are yellow. the cards are well shuffled. suppose that you randomly draw two cards, one at a time, with replacement. let g1 = first card is green and let g2 = second card is green. (a) draw a tree diagram of the situation. (b) find p(g1 and g2). (c) find p(at least one green). (d) find p(g2|g1). (e) are g1 and g2 independent events? explain why or why not.

Explanation:

Step1: Calculate total players

Total = 26 + 35 + 45 = 106

Step2: 1a: P(Shirt 1-33)

26/106 ≈ 0.2453

Step3: 1b: P(≤210 lbs)

33/106 ≈ 0.3113

Step4: 1c: P(1-33 AND ≤210)

21/106 ≈ 0.1981

Step5: 1d: P(1-33 OR ≤210)

(26+33-21)/106 = 38/106 ≈ 0.3585

Step6: 1e: P(1-33|≤210)

21/33 ≈ 0.6364

Step7: 2b: P(G1 AND G2)

(5/8)(5/8)=25/64≈0.3906

Step8: 2c: P(at least 1 green)

1 - (3/8)(3/8)=55/64≈0.8594

Step9: 2d: P(G2|G1)

5/8=0.625

Step10: 2e: Independence check

P(G2|G1)=P(G2), so independent

Answer:

1a. 0.2453
1b. 0.3113
1c. 0.1981
1d. 0.3585
1e. 0.6364
2b. 0.3906
2c. 0.8594
2d. 0.625
2e. Yes, because P(G2|G1) = P(G2) = 5/8.