A game of lucky dip consists of drawing 2 chips without replacement from a box containing 1 green chip, 2 blue chips and 40 white chips. You are paid according to the type of chips drawn. The green chip carries a prize of $20 and a blue chip 10. A white chip carries no prize. Find the expected winnings from one game.
I am a bit confused whether do i multiply or add the 2 expected winnings of the chips for each possible combination?
Suppose random variable X can take value $x_1$ with probability $p_1$, value $x_2$ with probability $p_2$, and so on, up to value $x_k$ with probability $p_k$. Then the expectation of this random variable X is defined as,
Here X is the winning from one game. X can take any value from ${0,10,20}$. Now $$P(X=20) = \frac 1 {43}$$ $$P(X=10) = \frac {2} {43}$$ $$P(X=0) = \frac {40} {43}$$
Hence the expected value of X (wiki), $$E(X) = 20P(X=20) + 10P(X=10) + 0P(X=0)$$