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The odds against John solving a problem are 4 to 3 and the odds in favor of Rafi solving the same problem are 7 to 5. What is the chance that the problem is solved when both them try it? - Mathematics and Statistics

Sum

The odds against John solving a problem are 4 to 3 and the odds in favor of Rafi solving the same problem are 7 to 5. What is the chance that the problem is solved when both them try it?

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Solution

Let J ≡ the event that John solves the problem

R ≡ the event that Rafi solves the problem

Odds against John solving the problem are 4 : 3

∴ P(J') = `4/(4 + 3) = 4/7`

∴ P(J) = 1 – P(J') = `3/7`

Odds in favour of Rafi solving the problem are 7 : 5

∴ P(R) = `7/(7 + 5) = 7/12`

∴ P(R') = 1 – P(R) = `5/12`

P(problem is solved) = 1 –  P(problem is not solved)

= 1 –  P(J' ∩ R')

= 1 –  P(J')·P(R')  ...[∵ events J' and R' are independent]

= `1 - 4/7*5/12`

= `1 - 5/21`

= `16/21`.

Concept: Odds (Ratio of Two Complementary Probabilities)
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