Question

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If $σ^2$ is the variance of X, then 100 σ2 is equal to ____.

Answer 1

X = Number of white ball drawn

$\begin{array}{l} P\left(X=0\right)=\frac{^6C_3}{^{10}C_3}=\frac{1}{6}\end{array}$

$\begin{array}{l} P\left(X=1\right)=\frac{^6C_2\times~^4C_1}{^{10}C_3}=\frac{1}{2} \end{array}$

$\begin{array}{l} P\left(X=2\right)=\frac{^6C_1\times~^4C_2}{^{10}C_3}=\frac{3}{10} \end{array}$

and $\begin{array}{l} P\left(X=3\right)=\frac{^6C_0\times~^4C_3}{^{10}C_3}=\frac{1}{30}\end{array}$

$\begin{array}{l} \text{Variance} = \sigma^2=\Sigma P_iX_i^2-\left(\Sigma P_iX_i\right)^2\end{array}$

\begin{array}{l} \sigma^2=\frac{1}{2}+\frac{12}{10}+\frac{3}{10}-\left(\frac{1}{2}+\frac{6}{10}+\frac{1}{10}\right)^2\end{array}$$\begin{array}{l} =\frac{56}{100}\end{array}

$100 σ^2 = 56.$