Chapter 12, Problem 24SGR

To determine

Find the expected value.

Answer to Problem 24SGR

  $6.53$

Explanation of Solution

Given information:

  $3$ nines, $4tens,5sixes,4fives,2twos$ and three are pulled from a deck of cards. If one card is drawn from the cards that were pulled out.

Calculation:

Probability of drawing $9$ ,

  $\begin{array}{l}P\left(X=9\right)=\frac{3}{19}\\ =0.157895\\ then,\exp ectedvalueis\\ x\times P\left(X=9\right)=9\times P\left(X=9\right)\\ =9\times 0.157895\\ =1.421053\end{array}$

Probability of drawing $10$ ,

  $\begin{array}{l}P\left(X=10\right)=\frac{4}{19}\\ =0.210526\\ then\\ x\times P\left(X=10\right)=10\times P\left(X=10\right)\\ =10\times 0.210526\\ =2.105263\end{array}$

Probability of drawing $6$ ,

  $\begin{array}{l}P\left(X=6\right)=\frac{5}{19}\\ =0.263158\\ then\\ x\times P\left(X=6\right)=6\times P\left(X=6\right)\\ =6\times 0.263158\\ =1.578947\end{array}$

Probability of drawing $5$ ,

  $\begin{array}{l}P\left(X=5\right)=\frac{4}{19}\\ =0.210526\\ then\\ x\times P\left(X=5\right)=5\times P\left(X=5\right)\\ =5\times 0.210526\\ =1.052632\end{array}$

Probability of drawing $2$ ,

  $\begin{array}{l}P\left(X=2\right)=\frac{2}{19}\\ =0.105263\\ then\exp ectedvalueis\\ x\times P\left(X=2\right)=2\times P\left(X=2\right)\\ =2\times 0.105263\\ =0.210526\end{array}$

Probability of drawing $3$ ,

  $\begin{array}{l}P\left(X=3\right)=\frac{1}{19}\\ =0.052632\\ then\exp ectedvalueis,\\ x\times P\left(X=3\right)=3\times P\left(X=3\right)\\ =3\times 0.052632\\ =0.157895\end{array}$

Now table shows the probability and expected value,

  $\begin{array}{cccccccc}X& 9& 10& 6& 5& 2& 3& total\\ P\left(X=x\right)& 0.157895& 0.210526& 0.263158& 0.210526& 0.105263& 0.052632& 1\\ xP\left(X=x\right)& 1.421053& 2.105263& 1.578947& 1.052632& 0.210526& 0.157895& 6.526316\end{array}$

  $\begin{array}{l}he\exp ectedvalueis,\\ E\left(X\right)={\displaystyle \sum xP\left(X=x\right)}\\ =6.53\end{array}$

Hence, the expected value is, $6.53$ .