It is observed that it rains on 12 days out of 30 days. Find the probability that it it will rain at least 2 days of given week.

#### Solution

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 will rain on at least 2 days of given week)

= P(X ≥ 2) = 1 – P(X < 2)

= 1 – [P(X = 0) + P(X = 1)]

`= 1 - [""^7C_0 (2/5)^0 (3/5)^(7 - 0) + "^7C_1 (2/5)^1 (3/5)^(7 - 1)]`

`= 1 - [1(1)(3/7)^7 + 7(2/5)(3/5)^6]`

`= 1 - [3/5 + 14/5](3/5)^6`

`= 1 - (17/5)(729/(5)^6) = 1 - 12393/5^7`

`= 1 - 12393/78125 = 1 - 0.1586`

= 0.8414

Hence, the probability that it rains at least 2 days of given week = `1 - 12393/5^7` OR 0.8414