15.4 Binomial

Binomial distribution The discrete binomial distribution is defined to be the probability of achieving $h$ successes in a given $N$ events where each event has a given $\theta$ probability of success.

$\begin{eqnarray*} P(h|N,\theta) = \binom{h}{N} \theta^{h} (1-\theta)^{N-h} \end{eqnarray*}$

Three dotted probability curves for the number of heads in 30 flips under different coins: p=0.1 peaks near 3 heads (about 0.24), p=0.5 peaks at 15 heads (about 0.145), and p=0.8 peaks near 24 heads (about 0.18). The three bells barely overlap. — **Figure C.4.** Probability of getting $h$ heads in 30 flips given a possible unfair coin. One coin has $p=0.1$ , where the maximum is for 3 heads (or 1/10 of the 30 flips), but 2 heads is nearly as likely. Another has $p=0.5$ , and is the fair coin considered earlier with a maximum at 15 heads (or 1/2 of the 30 flips). Finally, another coin shown as $p=0.8$ where 24 heads (or 8/10 of the 30 flips) is maximum.

Although it may look like a Beta, the binomial distribution is used to find the best estimate for the number of successes, $h$ , given the number of events, $N$ , and the probability of the success of a single event, $\theta$ .

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.