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Question
Three persons A, B and C apply for a job a manager in a private company. Chances of their selection are in the ratio 1:2:4. The probability that A, B and C can introduce chances to increase the profits of a company are 0.8, 0.5 and 0.3 respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A.
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Solution
Let E1 = Person A gets the job
E2 = Person B gets the job
E3 = Person C gets the job
A = No change takes place
The changes of selection of A, B and C are in the ratio 1:2:4
Hence, P(E1) = `1/7`, P(E2) = `2/7`, P(E3) = `4/7`
Also, given `P(A/E_1) = 0.2 = 2/10, P(A/E_2) = 0.5 = 5/10`
And `P(A/E_3) = 0.7 = 7/10`
Required probability is `P(E_1/A) = (P(A/E_1).P(E_1))/(P(A/E_1).P(E_1) + P(A/E_2).P(E_2) + P(A/E_3).P(E_3))`
= `(2/10 xx 1/7)/(2/10 xx 1/7 + 5/10 xx 2/7 + 7/10 xx 4/7)`
= `(2/70)/(2/70 + 10/70 + 28/70)`
= `2/40`
= `1/20`
∴ If no change takes palace, the probability that it is due to appointment of person A is `1/20`.
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Given that P(A) = `square`, P(C) = `square`
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P(L) = P(A ∩ L) + P(C ∩ L) + P(T ∩ L)
`="P"("A")*"P"("L"//"A") + "P"("C")*"P"("L"//"C") + "P"("T")*"P"("L"//"T")`
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`= square + square + square`
`= square`
`"P"("C"//"L") = ("P"("L" ∩ "C"))/("P"("L"))`
= `("P"("C") * "P"("L"//"C"))/("P"("L"))`
`= (square * square)/square`
`= square`
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