Introduction to Binomial Probability
The Binomial Distribution is used when we have a fixed number of independent trials, each with only two possible outcomes (often called 'success' and 'failure').
The Formula
For a random variable $X$ representing the number of successes in $n$ trials with probability of success $p$:
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$
Where:
- $n = 5$ (number of births selected)
- $p = 0.6$ (probability of a male birth)
- $q = 1 - p = 0.4$ (probability of a female birth)
Solving the Problem
Part i: Probability that exactly 3 are males
Here, we set $k = 3$.
$$P(X = 3) = \binom{5}{3} (0.6)^3 (0.4)^{5-3}$$ $$P(X = 3) = 10 \times 0.216 \times 0.16$$ $$P(X = 3) = 0.3456$$
Part ii: Probability that more than 4 are males
'More than 4' in a selection of 5 means exactly 5 successes ($k = 5$).
$$P(X > 4) = P(X = 5)$$ $$P(X = 5) = \binom{5}{5} (0.6)^5 (0.4)^0$$ $$P(X = 5) = 1 \times 0.07776 \times 1$$ $$P(X = 5) = 0.07776$$
Key Intuition
The binomial distribution helps us predict the likelihood of specific outcomes in a series of events. By calculating these probabilities, we can understand the variability inherent in random samples.