The probability that a certain kind of component will survive a given test is 2 3 . The probability that at most 2 components out of 4 tested, will survive is

Step-by-step solution

Let X be the number of components surviving the test, following a binomial distribution with n = 4 and success probability p = 2 3 . To find P(X 2) , compute P(X = 0) + P(X = 1) + P(X = 2) using the binomial PMF P(X = k) = 4 k ( 2 3 )^k ( 1 3 )^ 4 - k . Evaluate each term: P(X = 0) = 4 0 ( 2 3 )^0 ( 1 3 )^4 = 1 3^4 P(X = 1) = 4 1 ( 2 3 )^1 ( 1 3 )^3 = 8 3^4 P(X = 2) = 4 2 ( 2 3 )^2 ( 1 3 )^2 = 24 3^4 The sum is P(X 2) = 1 + 8 + 24 3^4 = 33 3^4 . This matches option C .