Question

A certain animal species is equally likely to have blue eyes or brown eyes. In a sample of four animals, what is the probability they all have brown eyes? What is the entropy of this sample?

Answer 1

The probability that all four animals in the sample have brown eyes is 1/16. The entropy of this sample is 1.

Step-by-step explanation:

In this question, we are asked to find the probability that all four animals in the sample have brown eyes. Since each animal is equally likely to have blue or brown eyes, the probability of one animal having brown eyes is 1/2. Since each animal's eye color is independent of the others, we can multiply the probabilities together:

P(Brown eyes for 1st animal) × P(Brown eyes for 2nd animal) × P(Brown eyes for 3rd animal) × P(Brown eyes for 4th animal) = (1/2) × (1/2) × (1/2) × (1/2) = 1/16

So the probability that all four animals have brown eyes is 1/16.

The entropy of the sample can be calculated using the formula:

Entropy = -P(Brown eyes) × log2(P(Brown eyes)) - P(Blue eyes) × log2(P(Blue eyes))

Since there are only two possible eye colors, the entropy can be calculated as:

Entropy = -1/2 × log2(1/2) - 1/2 × log2(1/2) = 1