Five independent trials of a binomial experiment with probability of success p=0.7 and probability of failure q=0.3 are performed. Find the probability of each event.
At most two failures
Recall that the formula for the binomial probability is given by
C(n,r)prqn−r
If the event should have at most two failures, then it means that the event must have at least three successes. Thus, we have
P(At least three successes)=P(exactly 3 successes)+P(exactly 4 successes)+P(exactly 5 successes)=C(5,3)(0.7)3(0.3)5−3+C(5,4)(0.7)4(0.3)5−4+C(5,5)(0.7)5(0.3)5−5=0.3087+0.36015+0.16807=0.83692
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