To determine the expected value of playing this game, we should calculate the average amount that a player can expect to win or lose per game over a long period of time.
The game offers two outcomes: winning and losing. Let's consider the probabilities and associated amounts for each of these outcomes.
1. Winning the game has a probability of 1/20. If you win, you earn $54.
2. Losing the game has a probability of 19/20. If you lose, you lose $3.
To find the expected value (EV), we can use the formula:
\[ EV = (P_{win} \times A_{win}) - (P_{lose} \times A_{lose}) \]
Where:
- \( P_{win} \) is the probability of winning
- \( A_{win} \) is the amount won when winning
- \( P_{lose} \) is the probability of losing
- \( A_{lose} \) is the amount lost when losing
Let's calculate:
\[ EV = \left(\frac{1}{20} \times 54\right) - \left(\frac{19}{20} \times 3\right) \]
Now, compute for each part of the equation:
The expected value from winning is \( \frac{1}{20} \times 54 = 2.70 \).
The expected value from losing is \( \frac{19}{20} \times 3 = 2.85 \).
Combining these two:
\[ EV = 2.70 - 2.85 \]
\[ EV = -0.15 \]
Thus, the expected value of playing this game is $-0.15. This means that, on average, a player would expect to lose 15 cents per game in the long run.
The correct choice from the given options is thus:
-$0.15
