Probability and expected value
Quick-fire odds, dice and cards, with exact answers
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Four options for each question, and you pick the one nearest the exact answer
20 questions · 90s each · up to 30 minutes · negative marking
- Negative markingLose a mark for each wrong answer
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- No going backOnce you move on, a question is locked
Topics
- Dice
- Coins
- Cards and draws
- Conditional probability
- Expected value
- Combinatorics
Adds to each question's clock
Primer
PrimerExpected value and conditional probability
Expected value
Expected value is what a game pays on average per go, if you could play it many times. Multiply each outcome by its probability and add the results.
A game costs £2 and pays £9 if a fair die shows a 6. You gain £7 with probability 1/6 and lose £2 with probability 5/6, so the expected value is 1/6 × £7 − 5/6 × £2 = −£0.50 a game.
Expectations add, even when the parts depend on each other. The expected total of three dice is 3 × 3.5 = 10.5, and the expected number of letters that land in the right envelope is 1 however many letters there are.
The fair price of a game is its expected payout. With a choice to make, such as whether to roll again, compare the expected value of each option and take the larger.
Conditional probability
P(A | B) is the chance of A once you know B has happened. Keep only the outcomes where B happens, then count the ones where A happens too: P(A | B) = P(A and B) ÷ P(B).
Two dice show at least one 6. That leaves 11 of the 36 outcomes, and only one of them is a double 6, so the chance of a double 6 is 1/11, not 1/36.
For screening questions, picture a round number of cases. Say 1% of payments are fraudulent and a check flags 90% of fraudulent payments and 5% of genuine ones. Out of 10,000 payments, 90 fraudulent and 495 genuine payments are flagged, so a flagged payment is fraudulent 90 times in 585, about 15%.
Counting shortcuts
Without replacement, multiply the shrinking fractions: two aces in a row is 4/52 × 3/51 = 1/221. For at least one of something, work out 1 minus the chance of none.
When order matters, multiply the choices left at each step: first, second and third place from 10 entrants is 10 × 9 × 8 = 720. When it doesn't, divide by the number of ways to order what you picked: 720 ÷ 3! = 120, which is C(10, 3).
Negative marking
Marking is a game with an expected value too. A blind guess from four options scores 1/4 × 1 − 3/4 × 1 = −0.5 on average, so skip it. Rule out two options and a guess scores 0 on average, so it neither helps nor hurts.