You get to throw a die. If you roll a 3, you win $6. If you roll a 1 or a 5, you

win $3. If you roll an even number, you lose $6. Find your expectation for this
game.

Thanks a lot

To find the expectation for this game, we need to calculate the expected value or average value of the amount of money you win or lose.

To do this, we need to consider the probabilities of each outcome and the corresponding amount of money won or lost. Let's calculate it step by step:

1. Probability of rolling a 3: Since there is only one face with a 3 on a standard die, the probability of rolling a 3 is 1/6. If you roll a 3, you win $6.
Expected value for rolling a 3 = (Probability of rolling a 3) * (Amount won for rolling a 3) = (1/6) * $6 = $1.

2. Probability of rolling a 1 or a 5: There are two faces with a 1 or a 5 on a standard die (one face with a 1 and one face with a 5). So, the probability of rolling a 1 or a 5 is 2/6 or 1/3. If you roll a 1 or a 5, you win $3.
Expected value for rolling a 1 or a 5 = (Probability of rolling a 1 or a 5) * (Amount won for rolling a 1 or a 5) = (1/3) * $3 = $1.

3. Probability of rolling an even number: There are three faces with even numbers (2, 4, and 6) on a standard die. So, the probability of rolling an even number is 3/6 or 1/2. If you roll an even number, you lose $6.
Expected value for rolling an even number = (Probability of rolling an even number) * (Amount lost for rolling an even number) = (1/2) * (-$6) = -$3.

Now, let's calculate the overall expected value or average value of the amount of money you win or lose in this game:

Overall expected value = (Expected value for rolling a 3) + (Expected value for rolling a 1 or a 5) + (Expected value for rolling an even number)
= $1 + $1 + (-$3)
= $2 - $3
= -$1.

The expectation for this game is -$1, which means that, on average, you can expect to lose $1 every time you play this game.

You have a 1/6 chance of getting $6, 1/3 (2/6) chance of getting $3, but a 1/2 (3/6) chance of losing $6. I would expect to lose money. Your chances are equal for winning $12 or losing $18.

I hope this helps. Thanks for asking.