Chapter 6: Q.9 (page 354)

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

Expert verified

a)Thus the probability distribution is

$X (S)$	$- 1$	$2$
probability	$0.75$	$0.25$

b)The expected value is $- 0.25$

Step by step solution

The probability of winning the game is $0.25 .$

The amount of bet is $1 .$

The payoff for the game is $3$ .

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

$p a y o f f = 3 - 1 = 2$

The probability distribution is calculated using the information provided.

$\begin{array}{lrr} X (S) & - 1 & 2 \\ Probability & 1 - 0.25 & 0.75 \end{array}$

Probability of wining the game is $0.25 .$

Amount of bet is $1 .$

Payoff for the game is $$ 3$ .

The expected value can be calculated as:

$E (X) = \sum x \times P (x) = - 1 (0.75) + (2) (0.25) = - 0.25$

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