Prob question on dice rolling and expected value

Question

So let's say I roll 3 dice.

• If they show all the same number, I will EARN 10£
• If all numbers are different, I will LOSE 2£
• If they show two numbers equal and one different, I will EARN 5£

What would be my expected return per roll?

If we calculate the probabilities (if I am not wrong)

• 6/216 --> earn 10£
• 90/216 --> earn 5£
• 120/216 --> lose 2£

How would I proceed to calculate the expected value per roll?

EDIT

Would the EV be: (6/216)x10 + (90/216)x5 - (120/216)x2 = 1.25?

Answer 1

By definition for discrete random variable, the expected value is the sum of the product of possible values of the random variable with it's probability. Say, X can be $10$ with probability $\frac{1}{2}$, X can be $\left(-100\right)$ with probability $\frac{1}{4}$, and X can be $0$ with probability $\frac{1}{4}$. Then the expected value is $10 \cdot\frac{1}{2} + \left(-100\right)\cdot\frac{1}{4} + 0\cdot\frac{1}{4} = \left(-20\right)$

So, the expected value is $10 \cdot\frac{6}{216} + 5\cdot\frac{90}{216} + \left(-2\right)\cdot\frac{120}{216} = \frac{270}{216}$