In a large population of 10,000 lab mice, each mouse has an independent 0.0003 probability of carrying a rare genetic mutation.
(A) Can the # of mice with the mutation be approximated using the Poisson distribution? If so, find $\lambda$.

• A. $3$
• B. $7$
• C. The # of mice cannot be approximated using Poisson distribution

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the conditions for using the Poisson distribution. The Poisson distribution is a good approximation for a binomial distribution when the number of trials (n) is large, the probability of success (p) is small, and the product of n and p (mean, λ) is moderate. Here, n = 10,000 and p = 0.0003.
2. Step 2: Calculate the mean (λ) of the Poisson distribution. The mean of a Poisson distribution is given by λ = n × p. Substitute n = 10,000 and p = 0.0003 into the formula.
3. Step 3: Verify if the conditions for the Poisson approximation are satisfied. Check if n is large (10,000 is large), p is small (0.0003 is small), and λ = n × p is moderate (you will calculate this in the previous step).
4. Step 4: If the conditions are satisfied, conclude that the number of mice with the mutation can be approximated using the Poisson distribution. If not, state that the Poisson approximation is not valid.
5. Step 5: If the Poisson approximation is valid, the value of λ calculated in Step 2 represents the mean number of mice expected to carry the mutation in the population.