# 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. - Mathematics and Statistics

Sum

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

Concept: Binomial Distribution
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 8 Binomial Distribution
Miscellaneous exercise 8 | Q 15.2 | Page 255