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Question
There is a working women's hostel in a town, where 75% are from neighbouring town. The rest all are from the same town. 48% of women who hail from the same town are graduates and 83% of the women who have come from the neighboring town are also graduates. Find the probability that a woman selected at random is a graduate from the same town
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Solution
Let the total number of women be 100.
∴ n(S) = 100
Let event N: Women are from neighbouring town,
event W: Women are from same town and
event G: Women are graduates.
Number of women from neighbouring town,
n(N) = 75
Number of women from same town,
n(W) = 25
∴ P(N) = `("n"("N"))/("n"("S")) = 75/100` and
P(W) = `("n"("W"))/("n"("S")) = 25/100`
`"P"("G"//"N"), "P"("G"//"W")` represent probabilities that woman is graduate given that she is from neighbouring town or same town respectively.
∴ `"P"("G"//"N") = ("n"("G"//"N"))/("n"("S")) = 83/100` and
`"P"("G"//"W") = ("n"("G"//"W"))/("n"("S")) = 48/100`
By Bayes’ theorem, the probability that a women selected at random is a graduate from the same town, is given by
`"P"("W"//"G") = ("P"("W") * "P"("G"//"W"))/("P"("W")*"P"("G"//"W") + "P"("N")*"P"("G"//"N"))``
= `((25/100)*(48/100))/((25/100)*(48/100) + (75/100)*(83/100))`
= `(25 xx 48)/((25 xx 48) + (75 xx 83))`
= `48/(48 + 249)`
= `16/99`
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