The Binomial Distribution Topic Review summary

Key learning points

• $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.
• $E(X)=np$, $\\mathrm{Var}(X)=np(1-p)$.
• Conditions: fixed trials, two outcomes, constant $p$, independence.
• Cumulative probabilities via tables/calculator; watch $\\le$ vs $<$.
• Normal approximation when $n$ large with continuity correction (S2).

Worked example 1 — Mean

$B(20,0.3)$. Mean?

Working

$np$.

Worked example 2 — PMF

$P(X=2)$ for $B(5,0.4)$.

Working

Expand carefully.

Worked example 3 — P(X=k)=\\\n{k}p^k(1-p)^{n-k}

$P(X=k)=\\\\binom{n}{k}p^k(1-p)^{n-k}$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — E(X)=np, \\\Var(X)=np(1-p)

$E(X)=np$, $\\\\mathrm{Var}(X)=np(1-p)$.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.