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 at most 2 days of a week.

Answer 1

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 at most 2 days of a week)
= P(X ≤ 2)
= P(X = 0 or X = 1 or X = 2)
= P(X = 0) + P(X = 1) + P(X = 2) 

= `""^7"C"_0(1/3)^0 (2/3)^7 + ""^7"C"__1 (1/3)(2/3)^6 + ""^7"C"_2(1/3)^2 (2/3)^5`

= `(2^5)/(3^7)(4 + 14 + (7 xx 6 xx 5!)/(2 xx 1 xx 5!))`

= `(2^5)/(3^7)(18 + 21)`

= `39(2^5/3^7)`.