Expected-Value Games

Sit through enough trading interviews and you notice most of the puzzles are the same puzzle wearing different clothes. Someone describes a little game with dice, cards, or coins, and asks: what's it worth? Or: what would you pay to play? Underneath, they are always asking you to compute one number — the expected value, the long-run average payoff. Learn to find it fast and calmly and you have cracked the whole genre.

The recipe never changes. Write down every outcome the game can produce, attach a probability to each, multiply, and add:

$$E[X] = \sum_i p_i\, x_i .$$

Here $x_i$ is a possible payoff and $p_i$ is the chance of getting it. The sum runs over every distinct thing that can happen. That is the entire machine. The skill an interviewer is testing is whether you can set it up cleanly under mild pressure, not whether you know exotic math.

Worked example: paid the higher of two dice

You roll two fair dice and get paid, in dollars, the larger of the two faces. What is this game worth?

The payoff is the maximum of the two rolls, so the outcomes are the numbers 1 through 6. The trick is the probabilities. For the max to equal $k$, both dice must be $\le k$ (probability $(k/6)^2$) but not both $\le k-1$. That works out to $P(\max = k) = (2k-1)/36$. So the chances climb steadily: a max of 6 is far more likely than a max of 1, because 6 wins whenever either die shows a 6.

Now feed the probabilities into the recipe:

$$E[\max] = \frac{1(1) + 2(3) + 3(5) + 4(7) + 5(9) + 6(11)}{36} = \frac{161}{36} \approx 4.47 .$$

So the game is worth about $4.47. If someone offers to let you play for $4, take it all day; at $5, decline. And if an interviewer asks you to make a market on it, you now have the anchor — quote a bid a little under 4.47 and an ask a little over, which is exactly the job in Fair Value and Quoting.

The shortcut that saves you: linearity

When a game is a sum of pieces, you rarely need the full distribution. By linearity of expectation, $E[X + Y] = E[X] + E[Y]$ whether or not the pieces are independent. Asked for the expected total of two dice? Don't build the 11-outcome table for the sum — each die averages 3.5, so the total averages $3.5 + 3.5 = 7$. Asked for the expected number of sixes in ten rolls? Each roll contributes $1/6$, and there are ten, so $10 \times \tfrac16 = 1.67$. Breaking a messy quantity into simple indicator pieces and adding is the single most reused move in this genre.

Where the number stops being the answer

Expected value prices a game you'll play many times. For a one-shot bet it can mislead, because it says nothing about the spread of outcomes. A game with a huge but rare jackpot and frequent small losses can have a lovely EV and still be a bad idea if a bad run wipes you out — that gap between "good EV" and "good bet" is what Edge and Bet Sizing and Kelly sizing exist to close. A second trap is hidden information: if you're told you won the right to play, ask whether winning itself is bad news, the essence of the winner's curse.

The habit to drill: hear a game, silently list outcomes and probabilities, compute the weighted average, then ask the follow-up an interviewer loves — how wide is the market you'd quote, and how much would you bet?