Question

Keno is a favorite game in casinos, and similar games are popular in the states that operate lotteries. Balls numbered $1$to $80$are tumbled in a machine as the bets are placed, then $20$of the balls are chosen at random. Players select numbers by marking a card. The simplest of the many wagers available is “Mark $1$Number.” Your payoff is $3$on a bet if the number you select is one of those chosen. Because $20$of $80$numbers are chosen, your probability of winning is $20/80$, or $0.25$. Let $X=$the amount you gain on a single play of the game.

(a) Make a table that shows the probability distribution of $X$.

(b) Compute the expected value of $X$. Explain what this result means for the player

Short answer

a)Thus the probability distribution is

$X\left(S\right)$	$-1$	$2$
probability	$0.75$	$0.25$

b)The expected value is$-0.25$

Step-by-step solution

Part(a) Step 1: Given Information

The probability of winning the game is $0.25.$

The amount of bet is $1.$

The payoff for the game is $3$.

Part (b) Step 2: Calculation

The reward is $3$in this case. If the game is won, the payout reduces the stake. To put it another way,

$$\begin{gathered} payoff=3-1 \\ =2 \end{gathered}$$

The probability distribution is calculated using the information provided.

$\begin{array}{|lrr|}\hline \mathbf{X}\left(S\right)& -1& 2\\ \text{Probability}& 1-0.25& 0.75\\ \hline\end{array}$

Part (b) Step 1: Given Information 

Probability of wining the game is$0.25.$

Amount of bet is$1.$

Payoff for the game is$\$3$.

Part (b) Step 2: Calculation 

The expected value can be calculated as:

$$\begin{gathered} \mathrm{E}\left(X\right)=\sum x\times P\left(x\right) \\ =-1\left(0.75\right)+\left(2\right)\left(0.25\right) \\ =-0.25 \end{gathered}$$