Tom and Alice work independently in an attempt to solve a certain problem (i. e. whether one of them solves it does not affect the chances that the other will solve it). The probability that Tom solves the problem is 0.25 and the probability that Alice solves the problem is 0.4.

(a) What is the probability that the problem will be solved by at least one person?
(b) If you find out later that the problem has been solved by at least one person, what is the conditional probability that Tom solved it?

(a) 0.325

(b) 0.385

Step-by-step explanation:

(a) Here we have the probability that the problem is solved by at least one person is given as follows

Since there are two people, Tom and Alice then there is a 50% chance that where the problem is solved by at leas one person it is either Tom or Alice

Therefore;

P(Solved by at least one) = P(Solved by Tom) + P(Solved by Alice)

Where each have 50%, we have;

P(Solved by at least one) = P(solved | Tom) × P(Tom) + P(solved | Alice) × P(Alice)

P(Solved by at least one) = 0.25×0.5 + 0.4×0.5 = 0.325.

(b) The conditional probability that the problem has been solved and that Tom solved the it is given by

P Solved and solved by Tom = (Probability of Tom solving the problem × Probability that the problem is solved by one person out of 2) ÷ (Probability that the problem has been solved by at least one person)

P(Tom | Solved) = P(Tom and Solved by one) ÷ P(Solved)

P(Tom | Solved) = [(0.25)(1/2)] ÷ 0.325 = 5/13= 0.385.

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