X binomial with mean 8, variance 4.

Question

$X$ binomial with mean 8, variance 4. If $P(X\le2)=\dfrac{k}{2^{16}}$, find $k$.

Accepted Answer

$np=8,npq=4\Rightarrow q=p=\dfrac12,n=16$.

$$P(X\le2) = {}^{16}C_0\left(\frac12\right)^{16}+{}^{16}C_1\left(\frac12\right)^{16}+{}^{16}C_2\left(\frac12\right)^{16} = \frac{1+16+120}{2^{16}}=\frac{137}{2^{16}}$$

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