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Question
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on exactly 3 days of a week.
Solution
Let X denote the number of days it rains in a week.
P(it rains) = p = `(10)/(30) = (1)/(3)`
∴ q = 1 – p = `1 - (1)/(3) = (2)/(3)`
Given, n = 7
∴ X ~ B`(7, 1/3)`
The p.m.f. of X is given by
P(X = x) = `""^7"C"_x (1/3)^x (2/3)^(7 - x),x` = 0, 1, ...,7
P(it rains on exactly 3 days of a week)
= P(X = 3)
= `""^7"C"_3(1/3)^3 (2/3)^4`
= `(7!)/(3! xx 4!) xx (1)/(3^3) xx (2^4)/(3^4)`
= `(7 xx 6 xx 5 xx 4!)/(3 xx 2 xx 1 xx 4!) xx (16)/(3!)`
= `(35 xx 16)/(3^7)`
= `(560)/(3^7)`.
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