Understanding Binomial Probability
When we toss a coin, each trial is independent and has only two possible outcomes: Heads (H) or Tails (T). This is the classic setup for a Binomial Distribution. The probability of success (getting a head) is $p = 0.5$, and the probability of failure is $q = 1 - p = 0.5$.
The probability of getting exactly $k$ successes in $n$ trials is given by the formula: $$P(X = k) = \binom{n}{k} p^k q^{n-k}$$ Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
Part (i): Probability of exactly two heads
Here $n = 5$ and $k = 2$.
- Calculate combinations: $\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10$.
- Calculate probability: $P(X=2) = 10 \times (0.5)^2 \times (0.5)^3 = 10 \times (0.5)^5$.
- Result: $10 \times \frac{1}{32} = \frac{10}{32} = 0.3125$.
Part (ii): Probability of at least two heads
"At least two" means $X \ge 2$. It is easier to calculate the complement: $$P(X \ge 2) = 1 - [P(X=0) + P(X=1)]$$
- $P(X=0) = \binom{5}{0} (0.5)^5 = 1 \times 0.03125 = 0.03125$
- $P(X=1) = \binom{5}{1} (0.5)^5 = 5 \times 0.03125 = 0.15625$
Sum of complement: $0.03125 + 0.15625 = 0.1875$.
Final result: $1 - 0.1875 = 0.8125$.