Answer:
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)
Step-by-step 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)
