I'm calculating probabilities for a card game that I'm creating. In this game, there are 78 cards in the deck. The cards have the following values:
4 cards have value 1 4 cards have value 2 4 cards have value 3 4 cards have value 4 4 cards have value 5 20 cards have value 1.5 38 cards have value 0
If I draw one card from this deck of 78, the mean value of that draw is 1.13 ((4*1)+(4*2)+(4*3)+(4*4)+(4*5)+(20*1.5)+(38*0))/78.
What is the mean (i.e., expected) value of my hand if I draw two cards from the deck in a single draw? Or three, or four cards?
Let's say you draw $n$ cards and $X_i$ denotes the value of card $i$.
The rv's $X_i$ have equal distribution hence equal mean: let's say that $\mu:=\mathbb EX_1$.
$X:=X_1+\cdots+X_n$ is the total value of the hand with: $$\mathbb EX=\mathbb EX_1+\cdots+\mathbb EX_n=n\mu$$
on base of linearity of expectation.