We are working with a situation where we capture and release 12 fish with replacement from a lake where 22% of the fish are tagged. The variable $X$ represents the number of tagged fish captured.

Part a) Specify the distribution of $X$.

Since each fish is captured with replacement and the probability of capturing a tagged fish remains constant, we are dealing with a Binomial Distribution.

In this case:

• $n = 12$ (number of trials, i.e., the 12 fish captured)
• $p = 0.22$ (the probability of capturing a tagged fish, which is 22%)

Thus, $X$ follows the binomial distribution: $X \sim \text{Binomial}(n=12, p=0.22)$

Part b) What is the expected value of $X$?

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

Part c) What is the probability of getting 4 or more tagged fish?

We want to compute $P(X \geq 4)$ where $X \sim \text{Binomial}(12, 0.22)$.

The probability of getting 4 or more tagged fish can be computed as: $P(X \geq 4) = 1 - P(X < 4) = 1 - P(X \leq 3)$ To find $P(X \leq 3)$, we need to compute the cumulative probability for $X = 0, 1, 2, 3$.

Let me calculate that for you.

The probability of getting 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 $P(X \geq 4)$ is $0.261$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this problem:

1. What is the variance of the binomial distribution in this scenario?
2. What is the probability of capturing exactly 2 tagged fish?
3. How would the probability change if we captured 20 fish instead of 12?
4. What is the probability of capturing fewer than 2 tagged fish?
5. How does the binomial distribution compare to the normal distribution when $n$ is very large?

Tip: The sum of probabilities for all possible outcomes in a binomial distribution always equals 1. This helps verify calculations when working with probabilities.