3.3 Binomial Distribution

The distribution of the possible number of heads, given $N$ flips with a coin with probability $p$ of flipping heads, is referred to as the Binomial Distribution. It has the form of Equation 3.3, with the “fair coin” probability, $1/2$ , replaced with $p$ :

$\begin{eqnarray} P(h|N,p)&=&\frac{N!}{h!(N-h)!}\times p^{h}\times (1-p)^{N-h}\label{eq:binomial} \end{eqnarray}$

Probability of flipping $h$ heads and $t$ tails with an unfair coin Given the probability of flipping a single heads is, say, $p$ and the total number of flips is $N=h+t$ , we have the following equivalent forms:

$\begin{eqnarray} P(h,t)&=&\frac{(h+t)!}{h!t!}\times p^{h}\times (1-p)^{t}\\ \nonumber P(h,N)&=&\frac{N!}{h!(N-h)!}\times p^{h}\times (1-p)^{N-h} \\ \nonumber P(h,N)&=&\binom{N}{h}\times p^{h}\times (1-p)^{N-h} \end{eqnarray}$

where the probability of tails is $1-p$ .

Note

Probability of flipping $h$ heads and $t$ tails with an unfair coin

Given the probability of flipping a single heads as $p$ , and the total number of flips is $N=h+t$ , we have the following probability for $h$ heads and $t$ tails:

$\begin{eqnarray*} P(h,t)=\frac{(h+t)!}{h!t!}\times p^{h}\times (1-p)^{t} \end{eqnarray*}$

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 3.2.** 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.

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.