A game consists of rolling a die and pays off as follows: $5.00 for a 6, $3.00 for a 5, $2.00 for a 4, and no money otherwise. If it costs $2.00 to play, find the expected net winnings for a player of this game.
average winning = ( 5+3+2 )/6 = 10/6
net = 10/6 - 2 = -0.33
The house wins, in the end :)
To find the expected net winnings for a player in this game, we need to calculate the expected value.
The expected value (EV) is calculated by taking the sum of all possible outcomes multiplied by their probabilities.
In this case, we have four possible outcomes: rolling a 6, 5, 4, or any other number. Let's calculate the probability and payoff for each outcome:
1. Rolling a 6: The game pays $5.00 for rolling a 6. The probability of rolling a 6 on a fair six-sided die is 1/6.
2. Rolling a 5: The game pays $3.00 for rolling a 5. The probability of rolling a 5 is also 1/6.
3. Rolling a 4: The game pays $2.00 for rolling a 4. The probability of rolling a 4 is 1/6.
4. Any other number: The game does not pay anything for rolling any number from 1 to 3. The probability of rolling any number from 1 to 3 is 3/6 = 1/2. (Since there are three possible outcomes out of six total outcomes).
Now, let's calculate the expected value:
EV = (Probability of rolling a 6) * (Payoff for rolling a 6) + (Probability of rolling a 5) * (Payoff for rolling a 5) + (Probability of rolling a 4) * (Payoff for rolling a 4) + (Probability of any other number) * (Payoff for any other number)
EV = (1/6) * ($5.00) + (1/6) * ($3.00) + (1/6) * ($2.00) + (1/2) * ($0.00)
Simplifying the calculation, we get:
EV = $0.83 + $0.50 + $0.33 + $0.00
EV = $1.66
The expected net winnings for a player in this game is $1.66.
However, we also need to consider the cost of playing the game, which is $2.00. So, to find the expected net winnings, we subtract the cost from the expected value:
Expected net winnings = EV - Cost
Expected net winnings = $1.66 - $2.00
Expected net winnings = -$0.34
Therefore, the expected net winnings for a player in this game is -$0.34, which means on average, the player can expect to lose $0.34 per game.