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प्रश्न
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.
उत्तर
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
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