Probability
The six calculations behind most quant probability questions: counting outcomes, the shape of a sum, sequences of flips, waiting times, the price of a bet, and updating a belief when the alarm fires.
Quant interviews ask probability questions with numbers in them, and they reward a method over a memory. The method is the same six moves in different costumes: count the equally likely outcomes and decide whether order matters; see how sums pile up in the middle; use the complement whenever 'at least one' appears; wait 1 ÷ p tries for anything with probability p; price a bet by its expected value and judge it by its noise; and, when evidence arrives, multiply the prior odds by the likelihood ratio instead of trusting the accuracy figure.
Two habits from the old lesson survive here in numbers. State what is equally likely before counting anything, because the counting is only right if the outcomes are. And when a question conditions on something, write down which conditional is being asked for, because P(A | B) and P(B | A) are different numbers that people swap without noticing.
- Counting. n choose k, ordered against unordered, named people by symmetry, every pair, round tables.
- Dice sums. The triangle, 'at least' from the triangle or its complement, the higher of two dice, rolls for a face.
- Coins. Binomial counts, the complement, HT against HH, a bias from a run, the gambler's fallacy.
- Waiting. 1 ÷ p, still waiting after k, memorylessness, m successes, the coupon collector.
- Expected value. EV net of the fee, the standard deviation, fair probability and prize, n plays.
- Bayes. True and false alarms, both conditionals, the odds form, the false-alarm rate a target allows.
Write down what is equally likely, count it, and check which conditional you were asked for. If the question says 'at least one', take the complement. If it says 'how long until', the answer is 1 ÷ p and it does not shrink while you wait. If it says 'the test came back positive', ask for the base rate before anything else.
Six moves, in the order they build
Each step points to the chapter that practices it.
Chapters
Counting: does order matter?
10 minMost probability questions are counting questions in disguise. The one decision that matters is whether swapping two of the chosen things gives a different outcome.
- Count committees with n choose k and ordered picks with n × (n − 1) × …
- Find the chance a named person is chosen without counting anything
- Count every pair: round-robin matches, handshakes, and the number of teams from the number of matches
- Seat people at a round table by fixing one of them first
Two dice: why the total is a triangle
10 minEach face is equally likely, but the totals are not, because a total in the middle can be made in many ways and a total at the edge in only one. Count the ordered pairs.
- Count the ordered pairs that make a given total on two dice
- Read 'at least' questions off the triangle, or off its complement
- Find the expected value of the higher of two dice
- Solve for the number of rolls from the chance of seeing a face at least once
Coins: counts, complements and waiting for a pattern
12 minFlips have no memory, but sequences have structure. The number of heads is binomial; the wait for HT is shorter than the wait for HH; and a run of heads tells you nothing about the next flip, unless you doubt the coin.
- Compute the probability of exactly k heads in n flips
- Use the complement for 'at least one head'
- Explain why HT arrives before HH on a fair coin, and compute both waits
- Recover a coin's bias from the chance of an all-tails run
How long until it happens: 1 ÷ p
11 minIf something happens with probability p on each try, you expect to wait 1/p tries. The wait does not shrink while you wait, several successes take several waits, and collecting every type takes far longer than the count suggests.
- Compute the expected wait for a first success and the chance you are still waiting after k tries
- Explain memorylessness: why a long silence does not make the event 'due'
- Add waits for the m-th success
- Sum the coupon-collector stages and see why the last one dominates
Pricing a bet: expected value and the noise around it
11 minThe fair price of a bet is what it pays, weighted by how often. What that number does not tell you is how bumpy the road is, and how many plays it takes for the average to show up.
- Compute the expected value of a game net of its fee, and the edge
- Compute the standard deviation of one play, and why the fee does not appear in it
- Solve for the fair probability or the fair prize
- Scale the mean and the standard deviation over n plays and say what repetition does
The alarm fired. How worried should you be?
12 minA signal that is right 95% of the time can still be wrong most of the times it fires, because the thing it looks for is rare. Bayes' rule weighs the true alarms against the false ones.
- Count true and false alarms from a base rate, a sensitivity and a false-alarm rate
- Compute P(event | alarm) and P(event | silence)
- Use the odds form to find the cleanest false-alarm rate a target posterior allows
- Explain base-rate neglect and why doubling the base rate does not double the posterior