Compute exact probabilities for the binomial distribution including $P(X = k)$, $P(X \le k)$, $P(X \ge k)$, and use complementary counting

What this dot point is asking

NESA wants you to compute binomial probabilities of every type: exact $P(X = k)$, ranges $P(X \le k)$ or $P(X \ge k)$, and at-least/at-most using complementary counting.

The answer

Exact probability

For $X \sim B(n, p)$,

Cumulative probability (sum of pmf values)

For HSC problems with small $n$, sum the pmf values up to $k$.

Complementary probability

For "at least $k$":

For small $k$ (like $k = 1$ or $k = 2$), this is much faster than summing $k$ to $n$.

Standard problem patterns

Exact number of successes: $P(X = k)$ directly from the pmf.

At least one success: $P(X \ge 1) = 1 - P(X = 0) = 1 - (1 - p)^n$.

No successes: $P(X = 0) = (1 - p)^n$.

All successes: $P(X = n) = p^n$.

Between two values: $P(j \le X \le k) = \sum_{i = j}^{k} P(X = i)$.

Choosing the easier sum

If asked $P(X \ge k)$ and $n - k + 1$ is small, sum directly. If $n - k + 1$ is large, use $1 - P(X \le k - 1)$ (complementary).

For "at most $k$", consider whether $k$ or $n - k$ is smaller. If $k$ is smaller, sum directly.