	Suppose	22%	of	the	1,300	fish	in	a	lake	have	been	
tagged	as	part	of	a	study.	We	capture	and	release	12	fish,	with replacement.		
Let	X =	the	number	of	tagged	fish	we	capture.	Answer	the	following:
a) (6	points)	Specify	the	distribution	of	X:		X	~	________________________
b) (6	points)	What	is	the	expected	value	of	X? (Round	to	one	decimal place.)
c) (9	points)	What	is the probability	of	getting	4 or	more tagged	fish?		
That	is,	compute	P(X ≥	4).		Give	the	value	as	a	probability	(not	percentage)	
rounded	correctly	to	three	decimal	places.

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.

Suppose 22% of the 1,300 fish in a lake have been tagged as part of a study. We capture and release 12 fish, with replacement. Let X = the number of tagged fish we capture. Answer the following: a) Specify the distribution of X: X ~ ________________________ b) What is the expected value of X? (Round to one decimal place.) c) What is the probability of getting 4 or more tagged fish? That is, compute P(X ≥ 4). Give the value as a probability (not percentage) rounded correctly to three decimal places.

Solve a binomial distribution problem involving tagged fish in a lake. Learn to calculate the expected value and find the probability of capturing 4 or more tagged fish. This example covers key probability concepts, suitable for students in Grades 10-12.

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