Question

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?

Answer 1

(a) 0.325

(b) 0.385

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.

Answer 2

a) P ( At-least 1 solves correctly) = 0.55

b) P ( Tom / at-least 1) = 0.45454

Explanation:

Given:-

- Probability that Tom solves the problem, pt = 0.25

- Probability that Alice solves the problem, pa = 0.4

- Both probabilities are independent

Find:-

(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?

Solution:-

a) The probability for at-least one of them either "Tom or Alice" or " Both" solve the problem correctly. We can determine the required probability by subtracting the Probability that neither of them solve the problem correctly:

P ( At-least 1 solves correctly) = 1 - P ( Both answer incorrectly)

Where, the probability of both solving the questions incorrectly is:

P ( Both answer incorrectly) = ( 1 - pt)*( 1 - pa )

= ( 1 - 0.25)*(1 - 0.4 ) = 0.75*0.6

= 0.45

Hence, the required probability is:

P ( At-least 1 solves correctly) = 1 - 0.45

= 0.55

b) The conditional probability associated with Tom solves the question given that at-least one of them solved can be expressed as:

P ( Tom / at-least 1) = P ( onlyTom + Both solve) / P( at-least 1 solves)

- The probability that only Tom solves is Tom answers correctly and Alice answers incorrectly:

P ( Only Tom solves) = pt*(1-pa)

= 0.25*(0.6)

= 0.15

- The probability that Both solve is Tom answers correctly and Alice also answers correctly:

P ( Both answer correctly) = pa*pt

= 0.25*0.4

= 0.1

- The required conditional probability would be:

P ( Tom / at-least 1) = ( 0.1 + 0.15 ) / 0.55

= 0.25 / 0.55

= 0.45454