Numbers is a game where you bet $1.00 on any 2-digit number from 00 to 99. If your number
comes up, you get $80.00. Find the expected net winnings.
To find the expected net winnings, we need to calculate the probability of winning and losing and their corresponding payoffs.
In this game, there are a total of 100 possible outcomes since you can bet on any 2-digit number from 00 to 99.
The probability of winning is the number of winning outcomes divided by the total number of outcomes. In this case, there is only one winning outcome, the number you bet on, and the probability of winning is 1/100 or 0.01.
The payoff for winning is $80.00.
The probability of losing is the number of losing outcomes divided by the total number of outcomes. There are 99 losing outcomes (all the other numbers except the one you bet on), so the probability of losing is 99/100 or 0.99.
The payoff for losing is -$1.00 (you lose the dollar you bet).
Now, we can calculate the expected net winnings using the formula:
Expected Net Winnings = (Probability of Winning * Payoff for Winning) + (Probability of Losing * Payoff for Losing)
Expected Net Winnings = (0.01 * $80.00) + (0.99 * -$1.00)
Expected Net Winnings = $0.80 - $0.99
Expected Net Winnings = -$0.19
Therefore, the expected net winnings in this game is -$0.19, which means on average you would lose $0.19 per game.