Binomial Distribution Question?

Question

The variable T~B(n, 0.96) and it is given that P(T=n) > 0.5. Find the greatest possible value of n.

Answer 1

A binomial distribution refers to situation that has only two possible outcomes such as taking a test or flipping a coin, so it is about the probability of some sort of success or failure but repeated multiple times.

I think the question means this. If 0.96 is the probability of a single successful event, what is the greatest possible value of the number of events n, such that the combined probability of n good events remains above 5?

If we directly solve 0.96^n = 0.5 we get n ~ 16.97

n must be an integer, and 0.96^17 ~ 0.499587 which is just below 0.5

So, the greatest value of n is 16.

Answer 2

T ~ B(n,0.96) represents a binomial distribution with n independent trials and probability of success being 0.96.

You are looking for all n trials to be successful, hence:

P(T=n) = 0.96^n

And you want the probability of all n trials being successful to be higher than 0.5:

0.96^n > 0.5

One method would be to just try values of n until the result is below 0.5 and take the value just before that point. But that could be time-consuming.

A better approach is to just solve mathematically for n.

Start by taking the log of both sides:

log(0.96^n) > log(0.5)

n log(0.96) > log(0.5)

log(0.96) is negative, so we change the direction of the inequality symbol when we divide:

n < log(0.5)/log(0.96)

n < 16.9797480183

n represents the number of trials so must be whole number. The largest whole number less than 16.9797480183 is just 16 (round down).

Double-check:

0.96^16 ≈ 0.5204 <--

For completeness, we can show that the probability drops below 0.5 when you have 17 trials:

0.96^17 ≈ 0.4996

Answer:

n = 16