probability and statistics

Question

Find the probability of at least one failures in five trials of a binomial experiment in which the probability of success is 30%. Round to the nearest tenth of a percent

Answer 1

At least one failure means there were not 0 failures, which means all the trials were successes.

So find the probability of getting five successes in five trials, and subtract this quantity from 1.

Answer 2

If the probability of one success is 30%, then the probability of one failure is 70%. If we're trying to find the probability of at least one failure, then one approach is to find the probability of exactly 1, 2, 3, 4, and 5 failures. That's a lot of calculation.

Happily, there is an easier alternative. We can find the probability of this not happening and subtracting from 1. There is only one calculation for that, and that is 0 failures. If we make X the number of failures, then

P(X = 0) = ₅C₀ • 0.7⁰ • 0.3⁵ = 0.00243. This is the probability of not having at least one failure. So the probability of the opposite event--having at least one failure--is 1 - 0.00243 = 0.99757.