(Solution) - There are 50 000 gnats and each leaves with probability 0 037 Suppose -(2025 Original AI-Free Solution)

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There are 50,000 gnats, and each leaves with probability 0.037.
Suppose many gnats are flying around in a room. Each leaves independently with a probability that depends on the insect repellent tested. In each case:
a. Describe the random variable with a binomial distribution giving the probability that exactly k have left.
b. Find the Poisson distribution approximating the probability that exactly k have left and the expected number to leave.
c. Find the variance of the binomial random variable and the Poisson approximation.