Find the expected payback for the games of chance described...

Find the expected payback for the games of chance described in Exercises 47–52. Keno In one form of the game Keno, the house has a pot containing 80 balls, each marked with a different number from 1 to 80. You buy a ticket for $\$ 1$ and mark one of the 80 numbers on it. The house then selects 20 numbers at random. If your number is among the $20,$ you get $\$ 3.20$ (for a net winning of $\$ 2.20$ )

Find the expected payback for the games of chance described in Exercises 47–52.
Keno In one form of the game Keno, the house has a pot containing 80 balls, each marked with a different number from 1 to 80. You buy a ticket for $\$ 1$ and mark one of the 80 numbers on it. The house then selects 20 numbers at random. If your number is among the $20,$ you get $\$ 3.20$ (for a net winning of $\$ 2.20$ )

Key Concepts

Payback Percentage

Payback percentage is a metric used to express the ratio of the expected return relative to the amount wagered. It is calculated by dividing the expected winnings by the cost of playing, and it provides a clear indicator of how much of each dollar wagered is returned to the player on average, highlighting the fairness or attractiveness of a game.

Probability

Probability is the measure of the likelihood of an event occurring, and it is central to analyzing games of chance. By assessing the chance of winning based on the rules of a game, one can assign probabilities to different outcomes, which then become crucial in the calculation of expected value and overall expected payback.

Expected Value

Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random event when it is repeated many times. In games of chance, it helps quantify the average net gain or loss of a game by summing the products of each possible outcome with its corresponding probability, providing a measure of the payback or return a player can statistically expect over the long term.

Net Payoff

Net payoff refers to the financial outcome experienced after the cost of playing has been deducted from any winnings. In the context of games of chance, evaluating the net payoff for each potential outcome, whether a gain or a loss, is essential in constructing the overall expected value that determines how much a player might expect to earn or lose in the long run.

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