A family has four children. What is the probability that there are 2 girls and 2 boys?

• There are two approaches to solve this problem:
Method 1: Direct Counting
-Total Outcomes: Since each child can be a boy or a girl, there are 2 possibilities for each child, resulting in a total of 2 * 2 * 2 * 2 = 16 possible outcomes (combinations of genders for the 4 children).
-Favorable Outcomes: There is only 1 way to have exactly 2 girls and 2 boys (BBGG or GBGB).
-Probability: Probability = Favorable outcomes / Total outcomes = 1 / 16 = 1/16.
Method 2: Using Binomial Theorem This method utilizes the binomial theorem, which calculates the probability of getting a specific number of successes in independent trials.

• Formula:
-P (k successes) = nCk * p^k * (1 – p)^(n – k)

• where:
-n = total number of trials (children) = 4
-k = desired number of successes (girls) = 2
-p = probability of success (having a girl) = 1/2 (assuming equal probability for boys and girls)

• Calculation:
-P (2 girls) = 4C2 * (1/2)^2 * (1 – 1/2)^2
-= 6 * (1/4) * (1/4)
-= 1/16