Expected Value in Virginia’s Pick 3 Game In Virginia’s Pick 3 lottery game, you can pay \(1 to select a three-digit number from 000 through 999. If you select the same sequence of three digits that are drawn, you win and collect \)500.

a. How many different selections are possible?

b. What is the probability of winning?

c. If you win, what is your net profit?

d. Find the expected value.

e. If you bet \(1 in Virginia’s Pick 4 game, the expected value is -50¢. Which bet is better in the sense of a producing a higher expected value: A \)1 bet in the Virginia Pick 3 game or a $1 bet in the Virginia Pick 4 game?

Virginia’s Pick lottery involves selecting a three-digit number from 000 to 999. The price of the bet is equal to $1. If the lottery is won, $500 is collected.

a.

The total number of digits available is equal to 10.

The number of digits to be selected is equal to 3.

As the repetition of digits is allowed, each selected digit has 10 options.

Thus, the number of selections is equal to

$10\times 10\times 10=1000$

Therefore, the number of selections possible is equal to 1000.

b.

The total number of selections is equal to 1000.

The number of combinations of digits that will result in a win is equal to 1.

The probability of winning is computed below.

$$\begin{gathered} P\left(\text{winning}\right)=\frac{1}{1000} \\ =0.001 \end{gathered}$$

Therefore, the probability of winning is equal to 0.001.

c.

The net profit on a $1 bet is computed below.

$$\begin{gathered} \text{Net}\text{profit}=\text{Amount}\text{won}-\text{Bet}\text{amount} \\ =500-1 \\ =499\text{dollars} \end{gathered}$$

Thus, the net profit is equal to 499 dollars.

d.

The expected value of the bet is equal to the expected amount that can be won or lost.

It is computed as follows.

$$\begin{gathered} \text{Expected}\text{value}=\left(\text{Net}\text{profit}\right)\times \left(\text{Probability} \\ \text{of}\text{winning}\right)-\left(\text{Net}\text{loss}\right)\times \left(\text{Probability} \\ \text{of}\text{losing}\right) \\ =\left(500-1\right)\left(0.001\right)-\left(1\right)\left(1-0.001\right) \\ =-0.50\text{dollars} \\ =-50\text{cents} \end{gathered}$$

Thus, the expected value of a $1 bet on Virginia’s Pick game is equal to -50 cents.

e.

The game that has a higher expected value is considered beneficial.

The expected value of betting on the Virginia Pick 4 game is equal to -50 cents.

As the expected value of betting on the Virginia Pick 3 game is equal to the expected value of betting on the Virginia Pick 4 game, both are equally suitable.