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%.