The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is ____________.

Solution

Maths sir · 4 steps

Mean and variance are two numbers with a known sum and product, so they are the roots of a quadratic.

1. Build the quadratic. Let the mean be np and the variance be npq. Their sum is 82.5 and their product is 1350, so both are roots of t² - 82.5t + 1350 = 0.

   t = (82.5 ± √(82.5² - 4 × 1350))/2 = (82.5 ± √1406.25)/2 = (82.5 ± 37.5)/2

2. Pick which is which. The roots are 60 and 22.5. Variance npq is mean np times q, and q < 1, so the variance is the smaller one: np = 60 and npq = 22.5.
3. Find q and p. Divide the variance by the mean: q = 22.5/60 = 3/8. So p = 1 - 3/8 = 5/8.
4. So. n = np/p = 60 × 8/5 = 96. The number of trials is 96.

The common slip: 160. Dividing the mean by q = 3/8 instead of p = 5/8 gives 160, not 96. Remember n = np/p.

The idea to keep. In a binomial distribution the variance is always less than the mean, because variance = mean × q. Their ratio gives q straight away, and then n = mean/p.

Answer from the NTA final answer key. Solution by PrepWiser.