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Question
A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, problem is solved?
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Solution
Let A be the event that student A can solve the problem.
B be the event that student B can solve problem.
C be the event that student C can solve problem.
∴ P(A) = `1/3`, P(B) = `1/4` and P(C) = `1/5`
P(A') = 1 − P(A) = `1-1/3=2/3`
P(B') = 1 − P(B) = `1-1/4=3/4`
P(C') = 1 − P(C) = `1-1/5=4/5`
Since A, B, C are independent events
∴ A', B', C' are also independent events
Let X be the event that problem is solved.
Problem can be solved if at least one of the three students solves the problem.
P(X) = P (at least one student solves the problem)
= 1 – P .........(no student solved problem)
= 1 – P (A' ∩ B' ∩ C')
= 1 – P(A') P(B') P(C')
= `1 - 2/3xx3/4 xx4/5`
= `1 - 2/5`
= `3/5`
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