Question

You agree to play this game in which the dealer tosses a die. You lose $2 if you get a one, two, or three. You lose $5 if you get a four or a five. But you win $20 if you get a six. Would you play the game? Answer this by computing the expected value per game.

Answer 1

The expected value of the game is $0.66, meaning you would expect to win 66 cents on average per game due to the fair distribution of outcomes. Therefore, based on the positive expected value, it could be beneficial to play the game.

Step-by-step explanation:

To determine whether you should play the game, you need to calculate the expected value, which is the average amount you can expect to win or lose per game if you played the game an infinite number of times.

Calculating the Expected Valu

The outcomes of rolling a die are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6 since it is a fair die.

• If you roll a 1, 2, or 3, you lose $2. The combined probability is 3/6 or 1/2.
• If you roll a 4 or 5, you lose $5. The combined probability is 2/6 or 1/3.
• If you roll a 6, you win $20. The probability is 1/6.

Using these probabilities and outcomes, we can compute the expected value (EV):

1. E(loss from rolling 1, 2, or 3): (-$2) x (1/2) = -$1
2. E(loss from rolling 4 or 5): (-$5) x (1/3) = -$1.67
3. E(win from rolling 6): $20 x (1/6) = $3.33

Add up these expected values to find the overall EV:

EV = -$1 - $1.67 + $3.33 = $0.66

Since the expected value is positive ($0.66), on average, you would expect to win 66 cents per game.

Conclusion

Based on the positive expected value, in theory, playing the game could be beneficial. However, it's important to bear in mind that this is an average calculated over many games, and it does not mean you will win every single game. There is still risk involved, and you could experience losses, especially in the short term.