Question

34. Find each of the following probabilities when n indepen- dent Bernoulli trials are carried out with probability of success p. a) the probability of no successes b) the probability of at least one success c) the probability of at most one success d) the probability of at least two successes

Answer 1

A.) (1 - p)^n

B.) 1 - (1 - p)^n

C.) (1 - p)^n + np*(1-p)^(n-1)

D.) 1 - (1 - p)^n - np*(1-p)^(n-1)

Explanation:

General form of a binomial probability :

P(x = x) = nCx * p^x * q^(n-x)

q = 1 - p ; n = number of trials ; x = number of successes ; p = probability of success

A.) probability of no successes ;

P(x = 0) = nC0 * p^0 * (1 - p)^(n-0)

P(x = 0) = 1 * 1 * (1 - p)^n

P(x = 0) = (1 - p)^n

Probability of atleast one success = 1 - P(no success)

P(x ≥ 1) = 1 - P(x = 0)

P(x = 0) = (1 - p)^n

P(x ≥ 1) = 1 - P(x = 0) = 1 - (1 - p)^n

Probability of at most one success

P(x ≤ 1) = p(x = 0) + p(x = 1)

P(x = 0) = (1 - p)^n

P(x = 1) = nC1 * p^1 * (1 - p)^(n-1)

P(x = 1) = n * p * (1 - p)^(n-1) = np*(1-p)^(n-1)

P(x ≤ 1) = (1 - p)^n + np*(1-p)^(n-1)

Probability of atleast two successes:

(1 - probability of at most 2 successes)

P(x ≥ 2) = 1 - P(x ≤ 1)

P(x ≥ 2) = 1 - (p(x = 0) + p(x = 1))

P(x ≥ 2) = 1 - p(x = 0) - p(x = 1))

P(x ≥ 2) = 1 - (1 - p)^n - np*(1-p)^(n-1)