Let’s solve each part of the problem step-by-step.

Given:

• Total fish in the lake = 1,300
• Tagged fish = 22% of 1,300 = $0.22 \times 1300 = 286$ tagged fish.
• Total sample size = 12 fish captured, with replacement.
• Let $X$ = the number of tagged fish captured.

a) Distribution of $X$:

Since we are capturing fish with replacement and the probability of getting a tagged fish remains the same for each capture, the number of tagged fish captured follows a binomial distribution.

The binomial distribution is defined as: $X \sim \text{Binomial}(n, p)$ where:

• $n = 12$ is the number of trials (fish captures),
• $p = \frac{286}{1300} = 0.22$ is the probability of capturing a tagged fish on each trial.

Thus, the distribution of $X$ is: $X \sim \text{Binomial}(12, 0.22)$

b) Expected value of $X$:

The expected value of a binomial distribution is given by the formula: $E(X) = n \cdot p$ Substituting $n = 12$ and $p = 0.22$: $E(X) = 12 \times 0.22 = 2.64$ Thus, the expected value of $X$ is 2.6 (rounded to one decimal place).

c) Probability of getting 4 or more tagged fish:

We want to compute $P(X \geq 4)$ for $X \sim \text{Binomial}(12, 0.22)$. This can be expressed as: $P(X \geq 4) = 1 - P(X \leq 3)$ We need to compute $P(X \leq 3)$ for a binomial distribution with $n = 12$ and $p = 0.22$.

To calculate this, we sum the probabilities for $X = 0$, $X = 1$, $X = 2$, and $X = 3$ using the binomial probability mass function (PMF): $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ We can compute this either by using a binomial table or a calculator.

Let me calculate this for you.

The probability of capturing 4 or more tagged fish, $P(X \geq 4)$, is approximately 0.261 (rounded to three decimal places).

Summary:

a) $X \sim \text{Binomial}(12, 0.22)$
b) The expected value of $X$ is 2.6.
c) The probability of getting 4 or more tagged fish is 0.261.

Do you want any further details or have any questions?

Hereare five related questions to expand on this problem:

1. How would the probability change if you captured the fish without replacement?
2. What is the variance of the distribution $X$?
3. What is the probability of capturing exactly 2 tagged fish?
4. How does the probability distribution change if we increase the number of captures to 20 instead of 12?
5. How does the expected value of $X$ change if the percentage of tagged fish increases to 30%?

Tip: For problems involving binomial distributions, always ensure whether you're sampling with or without replacement. Sampling without replacement often leads to a hypergeometric distribution instead.