Question

In a deer population, the male-to-female ratio is 1 : 2. The probability that a randomly formed group of size three has 2 males and 1 female is ________ (Round off to two decimal places).

Hint:

For problems involving a fixed number of successes in a given number of trials (like selecting males and females), use the binomial distribution to calculate the probability.

Answer 1

Correct Answer: 0.21

Given the male-to-female ratio of 1 : 2, the probability of selecting a male is: \[ P(\text{Male}) = \frac{1}{3}, \quad P(\text{Female}) = \frac{2}{3}. \] The problem asks for the probability of forming a group of 3 with exactly 2 males and 1 female. This is a binomial probability problem, where we calculate the probability for 2 males and 1 female in 3 selections. The formula is: \[ P(\text{2 Males, 1 Female}) = \binom{3}{2} \left( \frac{1}{3} \right)^2 \left( \frac{2}{3} \right)^1. \] The binomial coefficient \( \binom{3}{2} \) is 3, so: \[ P(\text{2 Males, 1 Female}) = 3 \times \left( \frac{1}{3} \right)^2 \times \frac{2}{3} = 3 \times \frac{1}{9} \times \frac{2}{3} = \frac{6}{27} = 0.222. \] Thus, the probability is: \[ \boxed{0.22}. \]