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Question
In a large school, 80% of the pupil like Mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like Mathematics.
Calculate the probabilities of obtaining an answer yes from 0, 1, 2, 3, 4 of the pupils.
Solution
Let X = number of pupils like Mathematics.
p = probability that pupils like Mathematics
∴ p = 80% = `80/100 = 4/5`
and q = 1 - p = `1 - 4/5 = 1/5`
Given: n = 4
∴ X ~ B `(4, 4/5)`
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`
i.e. p(x) = `"^4C_x (4/5)^x (1/5)^(4 - x)` x = 0, 1, 2, 3, 4
The probabilities of obtaining an answer yes from 0, 1, 2, 3, 4 of pupils are P(X= 0), P(X = 1), P(X = 2), P(X = 3) and P(X = 4) respectively.
i.e. `"^4C_0 (4/5)^0 (1/5)^(4 - 0)`, `"^4C_1 (4/5)^1 (1/5)^(4 - 1)` , `"^4C_2 (4/5)^2 (1/5)^(4 - 2)`, `"^4C_3 (4/5)^3 (1/5)^(4 - 3)` and `"^4C_4 (4/5)^4 (1/5)^(4 - 4)`
i.e. `1 (1)(1/5)^4, 4(4/5)*(1/5)^3, (4 xx 3)/(1 xx 2) (16/25)(1/25), 4(64/125)(1/5) and 1 xx (4/5)^4 (1/5)^0`
i.e. `(1/5)^4, 16/5 (1/5)^3, 96/5^2 (1/5^2), 256/5^3 (1/5) and 256/5^4`
i.e. `1/5^4, 16/5^4, 96/5^4, 256/5^4, 256/5^4`
OR `1/625, 16/625, 96/625, 256/625 and 256/625`
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