Question

In a lake with a large population of fish, there are 4 blue fish for every red fish. Every fish has an equal probability of being caught. If you dip a net into this lake and pick up 4 individuals at random, the probability that you will get 2 fish of each colour is:

Hint:

For hypergeometric distribution problems, use the formula: \[ P(X = k) = \frac{\binom{N_b}{k} \binom{N_r}{n-k}}{\binom{N}{n}} \] where: \(N_b\) and \(N_r\) are the numbers of blue and red fish, respectively,
\(n\) is the number of fish picked,
\(k\) is the number of blue fish picked.
This formula is useful when the probability is affected by the outcomes of previous draws.

Answer 1

Step 1: Determine the total ratio of blue to red fish.
The ratio of blue to red fish is 4:1. This means for every 5 fish, 4 are blue and 1 is red.

Step 2: Set up the hypergeometric distribution.
We want to calculate the probability of getting 2 blue fish and 2 red fish when picking 4 fish randomly. The hypergeometric probability is given by:
\[ P(X = k) = \frac{\binom{N_b}{k} \binom{N_r}{n-k}}{\binom{N}{n}} \] Where:
\(N\) is the total number of fish (assumed large),
\(N_b\) is the number of blue fish,
\(N_r\) is the number of red fish,
\(n = 4\) is the number of fish picked,
\(k = 2\) is the number of blue fish picked.

Step 3: Apply the formula for this specific scenario.
Since the population is large and the ratio is 4:1, we assume probabilities without worrying about the finite population.
Let the total number of fish be \(N = 5x\), where \(N_b = 4x\) and \(N_r = x\). Choose \(x = 20\) for simplicity, so \(N = 100\), \(N_b = 80\), \(N_r = 20\).

Applying the hypergeometric formula: \[ P(\text{2 blue, 2 red}) = \frac{\binom{80}{2} \binom{20}{2}}{\binom{100}{4}} \] \[ = \frac{3160 \times 190}{3921225} \approx 0.1531 \]

Final Answer:
The probability is approximately: \( \boxed{0.153} \)