Chapter 10.4, Problem 51E

Solution

To explain: The deal also shows the sponsor’s hope.

The sponsors would want the audience to opt for C with an expected value of 1000 whereas it would be most profitable to opt for B with an expected value of 1600.

Given information:

Deal A: Pick an envelope from five in his hand. The envelopes contain a ten-dollar bill, a twenty-dollar bill, a fifty-dollar bill, a hundred-dollar bill, and a check for five thousand dollars.

Deal B: Choose one of three suitcases. Two are empty and the third contains 240 twenty-dollar bills.

Deal C: Take $1000 with no strings attached.

Concept Used:

Expected Value for a Binomial Distribution:

If the binomial random variable $X$ counts the number of successes in $n$ independent trials with probability of success $p$ , then $\mu =E(x)=np$ .

Calculation:

Deal A has 5 outcomes each of which is equally likely, that is each has a probability of $\frac{1}{5}$ .

Substitute $x=10,20,50,100,5000$ and $p=\frac{1}{5}$ in the formula for the expected value.

  $\begin{array}{c}E\left(X_A\right)=\frac{1}{5}\cdot 10+\frac{1}{5}\cdot 20+\frac{1}{5}\cdot 50+\frac{1}{5}\cdot 100+\frac{1}{5}\cdot 5000\\ =1036\end{array}$

Deal B has 3 outcomes each of which is equally likely, that is each has a probability of $\frac{1}{3}$ .

Substitute $x=0,0,\left(240\times 40\right)$ and $p=\frac{1}{3}$ in the formula for the expected value.

  $\begin{array}{c}E\left(X_A\right)=\frac{1}{3}\cdot 0+\frac{1}{3}\cdot 0+\frac{1}{3}\cdot \left(240\times 20\right)\\ =1600\end{array}$

Deal C has a guaranteed return of 1000 which is its expected value.

The lowest expected value is for C and the highest for B.

Therefore, the sponsors would want the audience to opt for C whereas it would be most profitable to opt for B.