The finalists in an essay competition are Lisa (L), Melina (M), Ben (B), Danny (D), Eric (E), and Joan (J). Consider these finalists to be a population of interest. The possible samples (without Replacement)of size three that can be obtained from this population of six finalists are as follows. $$\begin{array} { l }
\text { L,M,B L,M,D L,M,E L,M,J L,B,D L,B,E } \
\text { L,B,J L,D,E L,D,J L,E,J M,B,D M,B,E } \
\text { M,B,J M,D,E M,D,J M,E,J B,D,E B,D,J } \
\text { B,E,J D,E,J }
\end{array}$$
If a simple random sampling method is used to obtain a sample of three of the finalists, what are the chances of selecting Ben, Danny, and Joan?

Question

The finalists in an essay competition are Lisa (L), Melina (M), Ben (B), Danny (D), Eric (E), and Joan (J). Consider these finalists to be a population of interest. The possible samples (without Replacement)of size three that can be obtained from this population of six finalists are as follows. $$\begin{array} { l } 
\text { L,M,B L,M,D L,M,E L,M,J L,B,D L,B,E } \
\text { L,B,J L,D,E L,D,J L,E,J M,B,D M,B,E } \
\text { M,B,J M,D,E M,D,J M,E,J B,D,E B,D,J } \
\text { B,E,J D,E,J }
\end{array}$$ 
If a simple random sampling method is used to obtain a sample of three of the finalists, what are the chances of selecting Ben, Danny, and Joan?

Quizplus · Accepted Answer

The total number of possible samples of size three from a population of six is calculated using combinations (since the order does not matter in simple random sampling). This is given by $ \binom{6}{3} = \frac{6!}{3!(6-3)!} = 20 $. Among these, only one specific combination includes Ben, Danny, and Joan (B,D,J). Therefore, the probability of selecting this specific group is $ \frac{1}{20} $.

The Finalists in an Essay Competition Are Lisa (L), Melina