In a dice game if you roll a 2, 4, or 6 you get the value of the die. If you roll as 1, 3, or 5 you lose $5. What is the expected value of the game?
To find the expected value of the game, we need to calculate the average amount of money you would win or lose per game.
Let's break down the probabilities and outcomes:
- The probabilities of rolling a 2, 4, or 6 are all 1/6 because there are six possible outcomes (the numbers 1 to 6) on a fair six-sided die, and each number has an equal chance of being rolled.
- If you roll a 2, 4, or 6, you win the value of the die, so you would win $2, $4, or $6, respectively.
- The probabilities of rolling a 1, 3, or 5 are also 1/6 each.
- If you roll a 1, 3, or 5, you lose $5.
Now, let's calculate the expected value:
Expected value = (probability of winning) x (amount won) + (probability of losing) x (amount lost)
Expected value = (1/6) x ($2 + $4 + $6) + (1/6) x (-$5 - $5 - $5)
Expected value = (1/6) x ($12) + (1/6) x (-$15)
Expected value = $2 - $2.50
Expected value = -$0.50
Therefore, the expected value of the game is -$0.50. This means, on average, you would expect to lose $0.50 per game.