1 . A fair coin is tossed three times, and the events A and B are defined as sho

Question

1. A fair coin is tossed three times, and the events A and B are defined as shown below.

A: {At most one head is observed} B: {The number of heads observed is even}

Note: Zero is considered to be an even number.

Find P(A U B) using the additive rule. (Answer should be 0.875)

2. In a population of 1000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 18 would test positive. If an athlete tests positive for testosterone, use Bayes's Rule to find the probability that the athlete is really doping. (Answer should be 0.735)

Explanation / Answer

1) A - at most 1 head

B - 0 heads or 2 heads

A U B = 0 head or 1 head or 2 heads

P(A U B) = P(0 head) + P(1 head) + P(2 heads)

= (1/2)3 + 3C1x(1/2)3 + 3C2x(1/2)3

= 0.875

2) P(doping) 100/1000 = 0.1

P( test positive) 0.1x0.5 + 0.9x18/900 = 0.05+0.018 = 0.068

According to Baye's rule, P(A | B) = P(A and B) / P(B)

So, P(doping | tested positive)

= P(doping and test positive)/P(test positive)

= 0.05/0.068 = 0.735

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