Question

a box contains 13 balls numbered 1 through 13. If 4 balls are selected one at a time from the box, without replacement, what is the probability the largest number will be 9

Answer 1

The probability that the sample of 4 balls contains the 9 ball is found as follows:
\[P(9isInSample)=\frac{\left(\begin{matrix}12 \\ 3\end{matrix}\right)}{\left(\begin{matrix}13 \\ 4\end{matrix}\right)}=\frac{12!}{3!9!}\times \frac{4!9!}{13!}=\frac{4}{13}\]
The probability that the sample of 4 balls contains three balls from the balls numbered 1, 2, 3, 4, 5, 6 ,7 and 8 is found as follows:
\[P(3ballsFrom1Thru8)=\frac{\left(\begin{matrix}8 \\ 3\end{matrix}\right)\left(\begin{matrix}5 \\ 1\end{matrix}\right)}{\left(\begin{matrix}13 \\ 4\end{matrix}\right)}=\frac{8!4!9!}{3!4!13!}=?\]

When you have found the value of the second probability by completing the above calculation, the probability the largest number will be 9 is found by multiplying the two values of probability together.