Let's say that I take a standard 54 card deck, and jettison from it all cards that are not a 1, 2, 3, or 4. This gives me a sixteen card deck consisting of four 1's, four 2's, four 3's, and four 4's. If I want to calculate the expected value of this sixteen car deck, I simple take $(1 \cdot (1/4) + 2 \cdot (1/4) + 3 \cdot (1/4) + 4 \cdot (1/4))$ to get $2.5$.
Let's say I then add the two Jokers back into the deck and add two new rules:
- If I draw a Joker, then I immediately draw a second card and add 1 to its face value (i.e. Joker + 4 = 5)
- If I draw a Joker, and then for my second draw, I draw the other Joker, I then draw a third card and add 2 to its face value (i.e. Joker + Joker + 2 = 4).
How would I model the Expected value of this setup?
My assumption is to treat drawing the combined drawing of Joker(s) into non-Joker cards as a single event, which would mean to triple the possible outcomes of any given draw from sixteen to forty-eight (sixteen outcomes where I draw a number card, sixteen outcomes where I draw a single joker into a number card, and sixteen outcomes where I draw both Jokers into a number card). Then I just add together the probabilities of drawing each value from one (where I draw a single one card) to six (where I draw two Jokers into a four) to find the Expected Value.
If I do it that way, I get 2.95. But am I on the right track for the method of calculating this?
Your approach is a fine one. Another approach is to compute the expected value contributed by the jokers and add that to your $2.5$ from drawing the number card. You have $\frac {16}{18}$ chance of drawing no jokers, $\frac 2{18}\cdot \frac {16}{17}$ of drawing one joker and $\frac 2{18}\cdot \frac 1{17}$ of drawing two jokers. This gives an expected value of $\frac {2\cdot16\cdot 1+2\cdot 1\cdot 2}{18\cdot 17}=\frac 2{17} \approx 0.12$ from jokers. Add that to the $2.5$ you get from the number card and the expected value is $2.62$. I think you make an error somewhere in the calculation because $2.95$ is too high.