You play a game where you win $1.00 for each dot that turns up on a fair die (has 6 sides). You pay $3.50 to play the game (and the money is not returned). Find the expected (average) winnings for this game.
To find the expected winnings for this game, we need to calculate the average amount of money you can expect to win per game.
The first step is to determine the probability of each outcome. Since the die is fair, each side has an equal chance of turning up. In this case, the probability of getting a dot (1, 2, 3, 4, 5, or 6) is 1/6.
Next, we need to determine the amount of money won or lost for each outcome. If a dot turns up, you win $1.00. However, you also pay $3.50 to play the game, which is a loss of $3.50.
Now, we can calculate the expected winnings:
(Expected winnings from getting a dot) = (Probability of getting a dot) * (Amount won per dot) = (1/6) * $1.00 = $0.1667
(Expected loss from playing the game) = (Amount paid to play the game) = $3.50
(Expected winnings) = (Expected winnings from getting a dot) - (Expected loss from playing the game) = ($0.1667) - ($3.50) = -$3.3333
Therefore, the expected winnings for this game are -$3.3333. This means, on average, you can expect to lose $3.3333 per game.