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a pond contains 80 fish. twenty of the fish have tags. a sample of 20 fish are randomly selected from the pond and put in a tank separate from the pond. what is the probability that 3 of the fish in the sample have tags?

User NoelHunter
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The probability of finding exactly 3 tagged fish in a sample of 20 fish from the pond is approximately 0.0307, or about 3.07%.

To calculate the probability of exactly 3 tagged fish in a sample of 20 from a pond of 80 with 20 tags, we can use the hypergeometric distribution.

This accounts for the fact that we are sampling without replacement, so the likelihood of drawing a tagged fish decreases as we take more fish out.

Here's how to find the probability:

Identify the parameters:

k = number of successes (tagged fish) needed: k = 3

m = number of successes in the population (tagged fish): m = 20

n = number of failures in the population (untagged fish): n = 80 - 20 = 60

K = sample size: K = 20

Use the hypergeometric probability formula:

P(k) = (m choose k) * (n choose K-k) / (N choose K)

where:

(m choose k) is the binomial coefficient representing the number of ways to choose k successes from m

(n choose K-k) is the binomial coefficient representing the number of ways to choose K-k failures from n

(N choose K) is the binomial coefficient representing the number of ways to choose K items from N

Calculate the probability:

(m choose k) = (20 choose 3) = 1140

(n choose K-k) = (60 choose 17) = 14,316,588,600

(N choose K) = (80 choose 20) = 372,690,988,236

P(k) = (1140 * 14,316,588,600) / 372,690,988,236 ≈ 0.0307

Therefore, the probability of finding exactly 3 tagged fish in a sample of 20 fish from the pond is approximately 0.0307, or about 3.07%.

User Mithil
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