dpble/prep

How to solve Beat the Odds questions

Probability: dice, cards, coins, conditional, Bayes, expected value and gambler's ruin. Skip anything that needs more than a minute. Press 1 to 5 to choose an option.

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How to solve them

Dice

  1. Cue: "two dice", "sum", "at least one 6". Everything is counted out of 36 (two dice) or 6 (one die).
  2. Method: for "exactly s" count the ways. For "at least s" count the ways and add them. For "at least one 6 in n rolls" use the complement: 1 - (5/6)^n.
  3. Setup: ways to make each sum 2 to 12 are 1,2,3,4,5,6,5,4,3,2,1. Memorize that row.
  4. Mental math: 1/36 is about 0.028, 6/36 = 0.167, 12/36 = 0.333. (5/6)^n for n = 1 to 6 is 0.833, 0.694, 0.579, 0.482, 0.402, 0.335.
  5. Check: the answer is between 0 and 1, and a sum never has more than 6 ways.

Cards

  1. Cue: "draw 2 cards without replacement", "both".
  2. Method: multiply P(first) by P(second given the first). Each draw removes a card, so top and bottom both drop by 1.
  3. Setup: k/52 x (k-1)/51, where k is the number of matching cards.
  4. Mental math: cancel first. 13/52 = 1/4, 26/52 = 1/2, 4/52 = 1/13, 12/52 = 3/13. Second factor: 12/51 is about 0.235, 25/51 is about 0.49, 3/51 = 1/17 is about 0.059.
  5. Check: the second factor is slightly smaller than the first, so the result is a bit under (k/52)^2.

Binomial

  1. Cue: "flip n times", "exactly k heads", "at least k heads".
  2. Method: P = C(n,k) x p^k x (1-p)^(n-k). For a fair coin it is C(n,k) / 2^n. For "at least k", add the counts for k to n, then divide once.
  3. Setup: read C(n,k) off Pascal's triangle. Rows: n=4: 1 4 6 4 1. n=5: 1 5 10 10 5 1. n=6: 1 6 15 20 15 6 1. n=7: 1 7 21 35 35 21 7 1. n=8: 1 8 28 56 70 56 28 8 1.
  4. Mental math: 2^n is 4, 8, 16, 32, 64, 128, 256, 512, 1024. Then 20/64 = 0.3125 and 70/256 is about 0.273. For biased coins: 0.3^2 = 0.09, 0.7^3 = 0.343, 0.6^2 = 0.36.
  5. Check: the counts in a row add up to 2^n, and the middle k has the biggest probability.

Conditional

  1. Cue: "given that", "if you know that".
  2. Method: shrink the world to B, then count how much of A lies inside it. P(A|B) = |A and B| / |B|.
  3. Setup: list or count the outcomes where B is true. That count is your new denominator, not 36.
  4. Mental math: counts are small integers, so make a fraction and simplify. 1/11 is about 0.091, so 5/11 is about 0.455. 1/9 is about 0.111.
  5. Check: the denominator is |B|. The answer is higher than P(A) if B makes A more likely.

Bayes

  1. Cue: "test is positive", "probability they actually have it".
  2. Method: natural frequencies. Imagine 10,000 people. Count true positives and false positives. Answer = TP / (TP + FP).
  3. Setup: sick = 10,000 x prevalence. TP = sick x sensitivity. FP = healthy x false-positive rate.
  4. Mental math: a percent of 10,000 just moves the decimal. 1% = 100, 2% = 200, 5% = 500, 10% = 1000. Then round the denominator: 99/594 is about 99/600 = 0.165.
  5. Check: for a rare disease the answer is far below the test accuracy.

Ev

  1. Cue: "expected", "average profit".
  2. Method: EV = sum of (payoff x probability). If outcomes are equally likely, just average the payoffs.
  3. Setup: list the payoff for each outcome, add them, divide by the number of outcomes.
  4. Mental math: one die averages 3.5, two dice sum to 7. Max of two dice is 161/36 (about 4.47), min is 91/36 (about 2.53), average |a-b| is 35/18 (about 1.94). Add first, divide last.
  5. Check: the EV lies between the smallest and largest payoff.

Expected rolls

  1. Cue: "expected number of rolls or flips until".
  2. Method: waiting for an event with probability p takes 1/p on average. For "every face appears" (coupon collector), add the waiting time for each new face: n/n + n/(n-1) + ... + n/1. For patterns like HH, set up states and solve a small equation.
  3. Setup: coupon collector = n x H_n, where H_n = 1 + 1/2 + ... + 1/n.
  4. Mental math: H_3 = 1.83, H_4 = 2.08, H_5 = 2.28, H_6 = 2.45. So 6 x 2.45 = 14.7. Pattern anchors: HH takes 6, HT takes 4, HHH takes 14.
  5. Check: the answer is at least n for coupon collector, and at least 1/p for a single event.

Gamblers ruin

  1. Cue: "start with $i", "stop at 0 or $N", fair bets.
  2. Method: in a fair game, P(reach N) = i/N and the expected number of rounds = i x (N - i).
  3. Setup: identify i (what you have) and N (your target).
  4. Mental math: i/N is a simple fraction. 3/10 = 0.3, 3/8 = 0.375, 5/12 is about 0.417, 7/12 is about 0.583. For duration multiply the two gaps, like 3 x 7 = 21.
  5. Check: starting in the middle gives probability 1/2 and the longest game.

Birthday

  1. Cue: "at least two share a birthday". Any "at least one" question usually means use the complement.
  2. Method: P(match) = 1 - P(all different). P(all different) = 365/365 x 364/365 x ... x (365-n+1)/365, one factor per person.
  3. Mental math: P(all different) is about e^(-n(n-1)/730). Work out n(n-1) first, divide by 730, then take e to the minus that. e^-1 = 0.368, e^-2 = 0.135, e^-3 = 0.050, and e^-x is about 1 - x for small x.
  4. Example n = 40: 40 x 39 = 1560, 1560/730 is about 2.14. e^-2 = 0.135 and e^-0.14 is about 0.87, so 0.135 x 0.87 is about 0.118. Answer is about 1 - 0.118 = 0.88 (exact 0.891).
  5. Anchors to memorize: 23 people is 0.507, 30 is 0.706, 40 is 0.891, 50 is 0.970.

Monty

  1. Cue: "host opens doors", "switch".
  2. Method: your first pick stays at 1/n. The host's reveal moves all the remaining probability onto the other closed door.
  3. Setup: P(switch wins) = P(first pick was wrong) = (n-1)/n.
  4. Mental math: 2/3 is about 0.667, 4/5 = 0.8, 9/10 = 0.9.
  5. Check: switching always beats staying (1/n).

On the real test: 5 multiple-choice options, and the exact answer may not be listed. Estimate to 2 digits and pick the closest.

Worked examples

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Optiver · Da Vinci · SIG · IMC · Citadel Securities · Five Rings

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