Mean, Variance and a Worked Illustration of the Binomial Distribution

Two summary numbers describe a binomial distribution completely, without needing to list every individual probability:

The mean $np$ is simply the expected count of successes — if an experiment with a $40\%$ success chance is repeated $5$ times, the expected number of successes is $5 \times 0.4 = 2$. The variance $npq$ measures how spread out the actual count of successes tends to be around that mean; because $q = 1 - p \le 1$, the variance $npq$ is always less than or equal to the mean $np$ — a useful check on any answer computed.

These two shortcut formulas are not assumed on faith — they can always be verified directly from the definition of an expected value, $E(X) = \sum x\, P(x)$, by actually listing every $P(x)$ and summing. Doing this once, for a small $n$, is what builds the confidence to trust the shortcut formulas when $n$ is too large for a full listing to be practical.

Illustration. Take $n = 5$ trials with $p = 0.4$ (so $q = 0.6$). The full probability distribution, computed directly from the p.m.f., is:

(Values are rounded to 4 decimals; they add to $1.0000$, confirming the p.m.f. was applied correctly at every step.) …