23 Among those who have a meal that Jordan cooked all those

23. Among those who have a meal that Jordan cooked, all those who found it delicious will tell him so, as will 40% of those who did not find it delicious (out of politeness). If 76% of the guests tell Jordan that the meal was delicious, what percentage actually thought it was?

24. If P{A | B} = 0.4, P{B} = 0.8, and P{A} = 0.5, are the events A and B independent? Justify your answer.

25. The probability that a lab specimen contains high levels of contamination is 0.1. Five independent samples are checked. (a) What is the probability that none contains high levels of contamination? (b) What is the probability that at least one contains high levels of contamination?

Solution

23.

Let

D = the event where the person said it was delicious
A = the event that it is actually delicious

Note that

P(D) = P(A) P(D|A) + P(A\') P(D|A\')

As P(D|A) = 1 (sure event),

P(D|A\') = 0.40,

P(A\') = 1 - P(A),

and P(D) = 0.76, then

P(D) = P(A) P(D|A) + P(A\') P(D|A\')

0.76 = P(A) (1) + (1 - P(A)) 0.40

0.76 = 0.40 + 0.60 P(A)

0.36 = 0.60 P(A)

Thus,

P(A) = 0.60

Those who actually thought the meal was delicious is 60%. [ANSWER]

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