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Question
A problem in Mathematics is given to three students whose chances of solving it are `1/3, 1/4` and `1/5`. What is the probability that the problem is solved?
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Solution
Let A1 denote the event of that first student solves the problem.
A2 denote the event that second student solves the problem.
A3 denote the event that third student solves the problem.
Given P(A1) = `1/3`
P(A2) = `1/4`
P(A3) = `1/5`
We note that A1, A2, A3 are independent events.
The problem will be solved if atleast one of them
solves it we have to find P(A1 ∪ A2 ∪ A3)
Probability of at least one solves the problem = 1 – Probability of no one solving it
P(A1 ∪ A2 ∪ A3) = `1 - "P"(bar"A"_1 ∪ bar"A"_2 ∪ bar"A"_3)`
= `1 - "P"(bar"A"_1) * "P"("A"_2) * "P"("A"_3)`
A1, A2, A3 are independent then `bar"A"1, bar"A"_2, bar"A"_3` are also independent.
= 1 – [1 – p(A1)] [1 – P(A2)] [1 – P(A3)]
= `1 - (1 - 1/3) (1 - 1/4) (1 - 1/5)`
= `1 - (2/3) (3/4) (4/5)`
= `1 - 2/5`
= `(5 - 2)/5`
= `3/5
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