4. For the following problems, you can assume independence. (These problems are

Question

4. For the following problems, you can assume independence. (These problems are a variation/extension of problems 74 and 76 of the Chapter 8 Review in the textbook.) Suppose the probability of a player hitting a home run is 1/20. In 5 tries what is the probability that the player hits at least 4 home runs given the player hits at least 1 home run? a. b. The recovery rate from a flu is 0.9. If 4 people have the flu, what is the probability exactly 2 will recover from the flu given at least 2 will recover from the flu?

Explanation / Answer

a) p = 0.05

P(X>= 4 | X>= 1)

= P(X >=4 and X >= 1)/P(X >= 1)

now P(X >=4 and X >= 1) = P(X>= 4)

P(X >= 4) = P(X = 4) + P(X =5) = 5* (1/20)^4 * 19/20 + (1/20)^5

= 0.00003

P(X>=1) = 1 - P(X =0) = 1 - (19/20)^5

= 0.2262190625

hence required answer

= 0.00003/0.2262190625= 0.00013261

b)

p = 0.9 , n = 4

P(X = 2 | X>=2)

= P(X = 2) /P(X >=2)

P(X =2) = 4C2 * 0.9^2 * 0.1^2 = 6 * 0.9^2 * 0.1^2 =0.0486

P(X >= 2) = 0.9963

hence answer = 0.0486/0.9963= 0.04878

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