Answer:
a) P ( At-least 1 solves correctly) = 0.55
b) P ( Tom / at-least 1) = 0.45454
Step-by-step 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
