A game consists of you rolling one fair 6-sided die and observing the number that appears uppermost. If you roll a 2 or a 4, then you win $10. If you roll a 6, then you win $20. If you roll any other number, then you do not win any money. Find your expected net winnings for this game if it costs $2 to play.
(1/3)(10)+(1/6)(20)-2 = 14/3 = $4.67
To find the expected net winnings for this game, we need to calculate the probability of each outcome and multiply it by the corresponding winnings or losses.
First, let's calculate the probability of rolling each number on a fair 6-sided die:
- P(2) = P(4) = 1/6 (since there is only one 2 and one 4 out of six possible outcomes)
- P(6) = 1/6 (since there is only one 6 out of six possible outcomes)
- P(1) = P(3) = P(5) = 1/6 (since each of these numbers has one out of six possible outcomes)
Now, let's calculate the winnings and losses for each outcome:
- If you roll a 2 or a 4, you win $10. So the winnings for this outcome are 10.
- If you roll a 6, you win $20. So the winnings for this outcome are 20.
- If you roll any other number (1, 3, 5), you do not win any money. So the winnings for these outcomes are 0.
The net winnings for winning outcomes (2 and 4) after deducting the cost to play are:
- Net Winnings for a 2 = $10 - $2 = $8
- Net Winnings for a 4 = $10 - $2 = $8
The net winnings for the winning outcome (6) after deducting the cost to play is:
- Net Winnings for a 6 = $20 - $2 = $18
The net winnings for the losing outcomes (1, 3, and 5) after deducting the cost to play are:
- Net Winnings for a 1 = $0 - $2 = -$2
- Net Winnings for a 3 = $0 - $2 = -$2
- Net Winnings for a 5 = $0 - $2 = -$2
Now, let's calculate the expected net winnings by multiplying each outcome's probability by its corresponding net winnings:
Expected Net Winnings = P(2) * Net Winnings for a 2 + P(4) * Net Winnings for a 4 + P(6) * Net Winnings for a 6 + P(1) * Net Winnings for a 1 + P(3) * Net Winnings for a 3 + P(5) * Net Winnings for a 5
Plug in the values:
Expected Net Winnings = (1/6) * $8 + (1/6) * $8 + (1/6) * $18 + (1/6) * (-$2) + (1/6) * (-$2) + (1/6) * (-$2)
Simplify the equation:
Expected Net Winnings = $4/3 + $4/3 + $3 - $1/3 - $1/3 - $1/3
Combine like terms:
Expected Net Winnings = $8/3 + $3 - $1/3
Combine fractions:
Expected Net Winnings = ($8 + $9 - $1) / 3
Calculate the final answer:
Expected Net Winnings = $16/3
Therefore, your expected net winnings for this game, after deducting the cost to play, is $16/3 or approximately $5.33.