Question

It is observed that it rains on 12 days out of 30 days. Find the probability that it rains exactly 3 days of week.

Answer 1

Let X = number of days it rains in a week.

p = probability that it rains

∴ p = `12/30 = 2/5`

and q = 1 - p = `1 - 2/5 = 3/5`

Given: n = 7

∴ X ~ B `(7, 2/5)`

The p.m.f. of X is given by

P(X = x) = `"^nC_x  p^x  q^(n - x)`

i.e. p(x) = `"^7C_x  (2/5)^x  (3/5)^(7 - x)` x = 0, 1, 2, ...., 7

P(it rains exactly 3 days of week) = P(X = 3)

= p(3) = `"^7C_3  (2/5)^3  (3/5)^(7 - 3)`

`= (7 xx 6 xx 5)/(3 xx 2 xx 1) (8/125)(81/625)`

`= 35(8/125)(81/625) = (35 xx 8 xx 81)/5^7`

`= 22680/78125 = 0.2903`

Hence, the probability that it rains exactly 3 days of week = `35 xx 8 xx 81/5^7` OR 0.2903.