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.

Answer 1

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`