Suppose that 50 identical batteries are being tested. After 8 hours of continuous use, assume that a given battery is still operating with a probability of .70 and has failed with a probability of .30. What is the probability that between 25 and 30 batteries (inclusive) will last at least 8 hours?

Answer 1

Recall that if the following conditions are met, we can use a binomial distribution to model the situation:
1. there are only two possible outcomes (operating or not operating, i.e. a bernoulli experiment)
2. probability applies to all units observed, and does not change.
3. random events

If we use the binomial distribution, where p=0.7 (success) and q=0.3 (failure) then the probability of success of i batteries out of n=50 are given by:
nCi*p^i*q^(n-i)
where nCi = n!/((n-i)!i!)
So calculate the probabilities for i=20,21,22,23,24,25 and sum them to get the probability that 20-25 batteries will remain operational.
I get about 0.0024

Answer 2

Sorry, it should be for 25-30 operating batteries, I get about P(25-30)=0.084