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 calculate the expected value of the game, we need to multiply the value of each possible outcome by its probability, and then sum all the results.
In this dice game, we have 3 possible outcomes:
1. Rolling a 2, 4, or 6: This outcome has a probability of 1/2 (since half of the possible outcomes are 2, 4, or 6) and each outcome has a value equal to the number shown on the die. So the expected value for this outcome is (2 + 4 + 6) * (1/2) = 12/2 = $6.
2. Rolling a 1, 3, or 5: This outcome also has a probability of 1/2 and each outcome incurs a loss of $5. So the expected value for this outcome is (-5) * (1/2) = -$2.5.
Now we can calculate the overall expected value by summing the expected values of each outcome:
Expected value = ($6) + (-$2.5) = $3.5.
Therefore, the expected value of the game is $3.5.
To find the expected value of a game, we need to multiply each possible outcome by its probability and then sum them up.
In this dice game, there are two possible outcomes: winning and losing. Let's calculate the expected value step by step:
1. Calculate the probability of winning:
- There are three winning outcomes (rolling a 2, 4, or 6) out of six total outcomes (the numbers 1 to 6 on a dice).
- So the probability of winning is 3/6 or 1/2.
2. Calculate the value of winning:
- If you win, you get the value of the die, which can be 2, 4, or 6.
- So the value of winning is (2 + 4 + 6)/3 = 4.
3. Calculate the probability of losing:
- There are three losing outcomes (rolling a 1, 3, or 5) out of six total outcomes.
- So the probability of losing is also 3/6 or 1/2.
4. Calculate the value of losing:
- If you lose, you lose $5.
5. Calculate the expected value:
- Multiply the value of winning by the probability of winning: 4 * 1/2 = 2.
- Multiply the value of losing by the probability of losing: -5 * 1/2 = -2.5 (negative since it represents a loss).
- Add the two values together: 2 + (-2.5) = -0.5.
Therefore, the expected value of this dice game is -$0.50.