Binomial and geometric distributions

Parametric equations

The binomial distribution $\mathrm{B}(n,p)$: $n$ independent trials, success probability $p$ on each trial, and $X$ the number of successes, so $X\in \{0,1,\dots ,n\}$.

The geometric distribution $\mathrm{Geo}(p)$: success probability $p$ on each trial, and $X$ the number of failures before the first success, so $X\in \{0,1,2,\dots \}$.

This page uses the count of failures before the first success, so the geometric distribution starts at $0$. If $X$ is instead the number of trials until the first success, the values are $\{1,2,\dots \}$ and $E(X)=1/p$. Taiwan’s 108 curriculum and the 113 AST geometric distribution item use that second version.

Interaction

• Binomial and Geometric: switch the bars and the formulas together
• Success probability p: drag $p$ and watch the bar shape, $E(X)$, and $\mathrm{Var}(X)$
• Trials n: in the binomial view, change $n$ and watch how many bars there are and where the peak sits
• E(X): the mark stays on the chart, with the band from $E(X)-\sigma$ to $E(X)+\sigma$, so the two distributions can be compared for how tightly they sit

What to notice

• The binomial distribution stops at $n$, so the bars run only from $0$ to $n$. The geometric distribution can reach large values, and its right tail is longer.
• As $p$ increases, the binomial mean $E(X)=np$ grows in proportion to $p$. For this failures-before-first-success geometric distribution, $E(X)=(1-p)/p$ falls, and not in a straight line.
• Both use the same success probability $p$ on each trial. One counts successes in a fixed number of trials. The other counts the failures before the first success.

Related

• Binomial distribution to the normal distribution

Further reading

• Binomial distribution (Wikipedia)
• Geometric distribution (Wikipedia)