Quick questions on Binomial probability calculations: exact values, cumulative probabilities and complements

The single most useful move in these questions is to count the complement when it is shorter. Take a five-question multiple-choice test with four options each, guessed at random, so $X \sim B(5, 0.25)$, and suppose the question asks for $P(X \ge 2)$, the probability of getting at least two right. The three panels show why the complement wins.

$P(X = k)$ directly from the pmf.

$P(X \ge 1) = 1 - P(X = 0) = 1 - (1 - p)^n$.

$P(X = 0) = (1 - p)^n$.

$P(X \ge k) = 1 - P(X \le k - 1)$, not $1 - P(X \le k)$ (which would double-count $X = k$).

For large $n$ (say $n = 100$), $0.5^{100}$ is small but not zero. Many calculators handle this fine; some need scientific notation.