If you learn one technique for quant interviews, learn this one. It converts a large class of apparently hard counting problems into arithmetic.
The statement
For any random variables X and Y, whether or not they are independent:
E[X + Y] = E[X] + E[Y]
The "whether or not they are independent" is the whole point, and it genuinely surprises people. You do not need to understand how the variables interact. You do not need the joint distribution. You just add.
The technique: indicators
Almost every interview application follows the same shape.
You are asked for the expected number of something. Write that count as a sum of indicator variables - one per item, equal to 1 if the thing happens and 0 otherwise. Then the expectation of the count is the sum of the individual probabilities, because E of an indicator is just P.
Worked example
In a shuffled deck of 52 cards, what is the expected number of cards that end up in their original position?
Directly, this is horrible - the events are dependent, and the distribution of the count is messy.
With indicators: card i is in its original position with probability 1/52. There are 52 cards. So the expectation is 52 times 1/52 = 1.
One line, no casework. The dependence between cards never enters.
Another
Expected number of adjacent pairs of the same colour when you shuffle 26 red and 26 black cards?
There are 51 adjacent slots. Each is a same-colour pair with probability 25/51, since given the first card, 25 of the remaining 51 match it. Answer: 51 times 25/51 = 25.
Where it breaks
It does not apply to products: E[XY] equals E[X]E[Y] only when X and Y are independent.
It does not apply to variance: Var(X + Y) = Var(X) + Var(Y) requires uncorrelated variables. Interviewers test this distinction specifically, so do not let the "no independence needed" habit leak from expectation into variance.
Practise it
The coupon collector and birthday problem both fall to it, and it appears constantly across the expected value question set.