A student takes a quiz consisting of 5 multiple choice questions. Each question has 4 possible answers. If a student is guessing the answer at random and answer to different are independent, then the probability of atleast one correct answer is

Concept: 

If there can be only 2 cases of true and false, then:

Probability of r true cases out of n (≥ r) P(x = r) = ⁿC_r p^rq^(n-r) 

where p is the probability of case being true and q is probability of case being false

Note: p and q are ≤ 1

Calculation:

The probability of correct answer p = \(\rm 1\over 4\) = 0.25

The probability of wrong answer q = \(\rm 3\over 4\) = 0.75

Total questions n = 5

Probability of atleast one correct answer (x ≥ 1):

P(x ≥ 1) = 1 - P(x = 0)

⇒ P(x ≥ 1) = 1 - ⁵C₀ p⁰q⁵

⇒ P(x ≥ 1) = 1 - (0.75)⁵ 

⇒ P(x ≥ 1) = 1 - 0.2373

⇒ P(x ≥ 1) \(\boldsymbol{\rm \approx}\) 0.7627