5. Binomial Distribution & Discrete Random Variables
Poisson Distribution
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In a large population of 10,000 lab mice, each mouse has an independent 0.0003 probability of carrying a rare genetic mutation.
(C) Estimate the probability that less than 3 mice carry the mutation.
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Verified step by step guidance1
Step 1: Recognize that this problem involves a binomial distribution because each mouse has an independent probability of carrying the mutation, and there is a fixed number of trials (10,000 mice). The binomial distribution is defined as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.
Step 2: Since the probability of success (p = 0.0003) is very small and the number of trials (n = 10,000) is large, the binomial distribution can be approximated by a Poisson distribution. The Poisson distribution is defined as P(X = k) = (λ^k * e^(-λ)) / k!, where λ = n * p is the expected number of successes.
Step 3: Calculate the expected number of successes (λ) using λ = n * p. Substitute n = 10,000 and p = 0.0003 into the formula to find λ.
Step 4: Use the Poisson distribution formula to calculate the probabilities for X = 0, X = 1, and X = 2 (less than 3 mice carrying the mutation). Add these probabilities together to estimate the probability that less than 3 mice carry the mutation.
Step 5: Verify the calculation by ensuring the sum of probabilities for X = 0, X = 1, and X = 2 is consistent with the Poisson distribution properties. This will give the final probability estimate for the problem.
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