Question

Assume that the committee consists of 10 Republicans and 12 Democrats. if a subcommitte of 6 members needs to be formed, What is the probability that the new subcommittee will contain at most 4 democrats

Answer 1

\[P(0\ Democrats)=\frac{\left(\begin{matrix}12 \\ 0\end{matrix}\right) \left(\begin{matrix}10 \\ 6\end{matrix}\right)}{\left(\begin{matrix}22 \\ 6\end{matrix}\right)}\]
\[P(1\ Democrat)=\frac{\left(\begin{matrix}12 \\ 1\end{matrix}\right)\left(\begin{matrix}10 \\ 5\end{matrix}\right)}{\left(\begin{matrix}22 \\ 6\end{matrix}\right)}\]
\[P(2\ Democrats)=\frac{\left(\begin{matrix}12 \\ 2\end{matrix}\right)\left(\begin{matrix}10 \\ 4\end{matrix}\right)}{\left(\begin{matrix}22 \\ 6\end{matrix}\right)}\]
\[P(3\ Democrats)=\frac{\left(\begin{matrix}12 \\ 3\end{matrix}\right)\left(\begin{matrix}10 \\ 3\end{matrix}\right)}{\left(\begin{matrix}22 \\ 6\end{matrix}\right)}\]
\[P(4\ Democrats)=\frac{\left(\begin{matrix}12 \\ 4\end{matrix}\right)\left(\begin{matrix}10 \\ 2\end{matrix}\right)}{\left(\begin{matrix}22 \\ 6\end{matrix}\right)}\]
After calculating the above 5 values of probability, you need to add the 5 values to find the required value of probability.