A new game is being introduced at the Hard Rock Cafe. A ball is spun around a wheel until it comes to rest in one of many spots. Whatever is listed in that spot will be the player's winnings. If the wheel has 12 spots labeled $1, 16 spots labeled $2, and 9 spots labeled $10, how much should a player expect to win on average?
Round to the nearest cent.
35 spots, so
12/35 * $1 + 16/35 * $2 + 9/35 * $10 = $3.83
To find the expected winnings on average, we can calculate the weighted average of the winnings based on the number of spots and the value assigned to each spot on the wheel.
First, let's calculate the total number of spots on the wheel:
Total spots = 12 + 16 + 9 = 37 spots
Next, let's calculate the average winnings for each spot:
Average winnings for $1 spots = $1
Average winnings for $2 spots = $2
Average winnings for $10 spots = $10
Now, we can calculate the weighted average of the winnings:
Weighted average = (Number of spots * Average winnings) / Total spots
For $1 spots:
Weighted average for $1 spots = (12 * $1) / 37
For $2 spots:
Weighted average for $2 spots = (16 * $2) / 37
For $10 spots:
Weighted average for $10 spots = (9 * $10) / 37
Finally, let's calculate the overall expected winnings:
Expected winnings = Weighted average for $1 spots + Weighted average for $2 spots + Weighted average for $10 spots
Expected winnings = ($1 * 12/37) + ($2 * 16/37) + ($10 * 9/37)
Simplifying this equation will give us the average winnings on average.