Probability of Weighted Dice

A game uses a 12-sided die which is rolled once. Each of the faces are labelled 1-12. The dice is weighted so it is four times as likely to roll a 6, and twice as likely to roll an 8, 10 or 12.

The game costs $20 to play.

If an even number is rolled, the player wins 30 dollars.

If a multiple of 3 is rolled, the player wins 40 dollars.

If any other number is rolled, they player loses the bet.

1. What is the probability of winning:

   
   
   

   a. Exactly 30 dollars

   
   
   

   b. Exactly 40 dollars




2. What is the expected return for each roll of the die?

I have calculated that the probability of rolling a 6 is 2/9.

The probability of rolling an 8, 10 or 12 is 1/9 each.

I do not know how to find the probability of winning EXACTLY 30 and 40 dollars, along with the expected return per roll.

Thanks!