Question

Find the probability of guessing correctly at least seven out of ten answers in a ‘true’ or ‘false’ objective test.

Answer 1

Let X = number of true answers

p = probability of true answer

∴ p = `1/2` and q = 1 – p = `1 - 1/2 = 1/2`

Given: n = 10

∴ `X ∼ B(10, 1/2)`

The p.m.f. on X is given as:

P[X = x] = `""^nC_xp^xq^(n - x)`

i.e. p(x) = `""^10C_x(1/2)^x(1/2)^(10-x)`

= `""^10C_x(1/2)^10, x = 0, 1, 2, ..., 10`

P(at least 7 answers are correct)

= P[X ≥ 7] = P[X = 7] + P[X = 8] + P[X = 9] + P[X = 10]

= p(7) + p(8) + p(9) + p(10)

= `""^10C_7(1/2)^10 + ""^10C_8(1/2)^10 + ""^10C_9(1/2)^10 + ""^10C_10(1/2)^10`  

= `""^10C_3(1/2)^10 + ""^10C_2(1/2)^10 + ""^10C_1(1/2)^10 + ""^10C_0(1/2)^10` ...[∵ ⁿC_x = ⁿC_{n – x}]

= `[""^10C_3 + ""^10C_2 + ""^10C_1 + ""^10C_0]*(1/2)^10`

= `[(10 xx 9 xx 8)/(1 xx 2 xx 3) + (10 xx 9)/(1 xx 2) + 10 + 1](1/2)^10`

= `(120 + 45 + 11)/2^10`

= `176/1024`

= 0.1718

Hence, the probability that at least 7 of 10 answers in a ‘true’ or ‘false’ objective test are correct is 0.1718