EX 3.30 cola drinker

Each of six randomly selected cola drinkers is given a glass containing cola $S$ and one containing cola $F$. The glasses are identical in appearance except for a code on the bottom to identify the cola. Suppose there is actually no tendency among cola drinkers to prefer one cola to the other. Then $p=P(\text{a selected individual prefers S})=.5$so with $X=\text{the number among the six who prefer S}$ and $X\sim \mathrm{Bin}\left(6,.5\right).$

Thus $P\left(X=3\right)=b\left(3;6,.5\right)=\left(\begin{array}{l}6\\ 3\end{array}\right){\left(.5\right)}^3{\left(.5\right)}^3=20{\left(.5\right)}^6=.313$

The probability that at least three prefer $S$ is

$$\begin{array}{rl}P\left(3\le X\right)& =\sum _{x=3}^{6}b\left(x;6,.5\right)\\ & =\sum _{x=3}^6\left(\begin{array}{l}6\\ x\end{array}\right){\left(.5\right)}^x{\left(.5\right)}^{6-x}\\ & =.656\end{array}$$

and the probability that at most one prefers $S$ is $P\left(X\le 1\right)=\sum _{x=0}^{1}b\left(x;6,.5\right)=.109$