Question

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

Answer 1

An experiment succeeds thrice as often as it fails. 

⇒ p = P(getting success) = and 

q = P(getting failure) = Here, number of trials = n = 5 

By binomial distribution, we have

P(X = x) = ⁿC_xq^{n - x}p^x, x = 0, 1, 2, ....n

P(X = r) = ⁵C_r(3/4)^r(1/4)^{5 - r}

Now , P(getting at least 3 success) = P(X = 3) + P(X = 4) + P(X = 5)

= ⁵C₃(3/4)³(1/4)² + ⁵C₄(3/4)⁴(1/4)¹ + ⁵C₅(3/4)⁵

= (3/4)³[10 x 1/16 + 15 x 1/16 + 9 x 1/16] = 27/64 x 34/16 = 459/512