91Ó°ÊÓ

When dealing with probability, we're looking at the likelihood of a specific event happening. In the case of determining the expected number of girls in a family with three children, we assume that the chance of having a girl is the same as having a boy. This means both have a probability of 0.5, or 50%.Each child is an independent event, and the outcome (girl or boy) of one event doesn't impact the next. This assumption is critical as it allows us to use specific mathematical tools to find the solution effectively.

A binomial distribution helps us describe the number of successes in a fixed number of trials, where each trial has two possible outcomes (success or failure). Here, success is having a girl, and failure is having a boy, though those terms are used mathematically! Given our fixed number of trials (three children) and the equal probability for each outcome, we use the binomial formula for calculation:\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)where:

• \(n = 3\) (number of children)
• \(k\) is the number of girls
• \(p = 0.5\) (probability of having a girl)

This formula helps us find the probability for each possible number of girls in the family.We calculate the probabilities for 0, 1, 2, and 3 girls:

• \(P(X = 0) = \binom{3}{0} (0.5)^0 (0.5)^3 = 0.125 \)
• \(P(X = 1) = \binom{3}{1} (0.5)^1 (0.5)^2 = 0.375 \)
• \(P(X = 2) = \binom{3}{2} (0.5)^2 (0.5)^1 = 0.375 \)
• \(P(X = 3) = \binom{3}{3} (0.5)^3 (0.5)^0 = 0.125 \)

The expected value is a fundamental concept in probability that gives us the average outcome we can expect over many trials of an experiment. For a discrete random variable, it is calculated by summing the product of each outcome and its probability. Mathematically, it is represented as:\( E(X) = \text{Number of girls } \times \text{ Probability of that number} \)For our case:\(E(X) = 0 \times 0.125 + 1 \times 0.375 + 2 \times 0.375 + 3 \times 0.125 \)Upon calculating, we get:\(E(X) = 0 + 0.375 + 0.75 + 0.375 = 1.5 \)So, on average, a family with three children is expected to have 1.5 girls. This result, although it might seem abstract since you can't have half a girl, means that over many families, the average will tend towards this number.