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 that you will win over the long run.
First, let's calculate the probability of each outcome. Since the die is fair, each of the six numbers (dots) has an equal probability of appearing, which is 1/6.
Next, we can determine the amount of money you can win for each possible outcome. If a dot appears on the die, you win $1.00. If no dot appears, you win $0.00.
Now, we can calculate the expected winnings using the following formula:
Expected winnings = (Probability of each outcome) * (Amount of money won for each outcome)
Since there are six possible outcomes (dots), and each outcome has a probability of 1/6, the expected winnings will be:
Expected winnings = (1/6 * $1.00) + (1/6 * $1.00) + (1/6 * $1.00) + (1/6 * $1.00) + (1/6 * $1.00) + (1/6 * $1.00)
= $1.00/6 + $1.00/6 + $1.00/6 + $1.00/6 + $1.00/6 + $1.00/6
= $6.00/6
= $1.00
Therefore, the expected winnings for this game are $1.00.