Homework SourseFind each of the following probabilities when n independent
Find each of the following probabilities when n independent 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
(e) the probability of no failures
(f) the probability of at least one failure
(g) the probability of at most one failure
(h) the probability of at least two failures
Solution
Probability of success = p
probability of failure = 1-p
total trials = n
Hence,
(a) the probability of no successes
= (1-p)n
(b) the probability of at least one success
= 1 - probability of no success
= 1 - (1-p)n
(c) the probability of at most one success
= probability of no success + probability of one success
= nC0 p0 (1-p)n + nC1 p1 (1-p)n-1
(d) the probability of at least two successes
= 1 - probability of atmost one success
= 1 - [ nC0 p0 (1-p)n + nC1 p1 (1-p)n-1 ]
(e) the probability of no failures
= all successes
= pn
(f) the probability of at least one failure
= 1 - probability of all successes
= 1 - pn
(g) the probability of at most one failure
= probability of no failure + probability of one failure
= nCn pn (1-p)0 + nCn-1 pn-1 (1-p)1
(h) the probability of at least two failures
= 1 - probability of at most one failure
= 1 - [ nCn pn (1-p)0 + nCn-1 pn-1 (1-p)1 ]
Hope this helps
