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Dice, coins and cards

The chance of a 7 with two dice, of at least one six in four rolls, of two aces in a row — counted out exactly, and given as a fraction.

  • Probability
  • Untimed
  • Endless levels

How it plays

Each question is a chance you can count: an event on one die, heads among several coins, a total on two dice, a card from a deck, balls drawn from a bag without putting them back, and a chance given something you were told.

Answers are exact fractions in lowest terms — or percentages to 1 dp where the fractions grow unreadable — and a typed answer is a percentage within half a point. New kinds unlock as you climb; the dice grow more sides, the bags more balls, the tries more numerous.

The wrong options are the real slips: 1/11 for a total because there are eleven totals, adding chances for "at least one", counting a card that fits both parts twice, forgetting that a drawn ball is not put back, and ignoring what you were told.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes counting chances of these kinds quicker and surer. It claims nothing wider.

Learn

How to get better at dice, coins and cards: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. List what can happen, as outcomes that are equally likely: the faces of one die, the 36 ordered pairs of two dice, the 2^n sequences of n coins, the 52 cards.
  2. Count the outcomes that fit the question. Use a shortcut below where one exists; otherwise list them.
  3. Write the chance as fitting ÷ all, and put it in lowest terms — the options are always reduced, so 6/36 appears as 1/6.
  4. For “at least one” and for draws without putting back, do not count outcomes directly: use the complement, or multiply the chances step by step.

Techniques and tricks

  • Two-dice totals

    With two n-sided dice, a total t comes up n − |t − (n + 1)| ways out of n². On six-sided dice that is 6 − |t − 7|: 7 has 6 ways, 6 and 8 have 5, down to 2 and 12 with 1.

    e.g. Two 8-sided dice, total 9: 8 ways out of 64 = 1/8

  • Doubles and differences

    Two n-sided dice show a double n ways out of n², and two numbers that differ by k in 2(n − k) ways. Always count ordered pairs: 2 then 5 is not 5 then 2.

    e.g. Six-sided, differ by 2: 2 × 4 = 8 of 36 = 2/9

  • At least one: turn it round

    The chance of at least one hit in n tries is 1 minus the chance of none, and none is (chance of a miss)^n. Never add the chances.

    e.g. At least one six in 4 rolls: 1 − (5/6)^4 = 1 − 625/1,296 ≈ 51.8%

  • Coins: Pascal’s triangle

    Exactly k heads from n coins can fall C(n, k) ways out of 2^n. The rows of Pascal’s triangle give C(n, k): 1 4 6 4 1 for four coins, 1 5 10 10 5 1 for five.

    e.g. 5 coins, exactly 2 heads: 10/32 = 5/16

  • Cards: add, then take off the overlap

    For “this or that”, add the two counts and subtract the cards that are both, or they are counted twice.

    e.g. A heart or a king: 13 + 4 − 1 = 16, so 16/52 = 4/13

  • Bags: one fewer each time

    Without putting back, multiply the chances draw by draw; after each draw there is one fewer of that colour and one fewer ball.

    e.g. 5 red, 3 blue, two reds: 5/8 × 4/7 = 20/56 = 5/14

  • Given that: shrink the list

    When you are told something, throw away every outcome that does not fit it and count again among the rest, which are still equally likely.

    e.g. Two dice, told the total is 10 or more (6 pairs): a double is 5-5 or 6-6, so 2/6 = 1/3

As the levels climb

  • You start with one die and coins. Then two dice, “at least one” and single cards arrive together; then draws from a bag without putting back; and last “Given that…”, which reuses two dice with a condition.
  • Between each new kind the numbers grow: dice with 8, 10, 12 or 20 sides, and later more; more coins, up to 16; more tries for “at least one”; bags with more balls and, later, three drawn instead of two.
  • Big dice and many coins make listing impossible, so lean on the formulas: n − |t − (n + 1)| for totals, C(n, k) for coins, 1 − miss^n for at least one.
  • “At least one” answers are percentages to 1 dp, not fractions, because the fractions grow unreadable. Work out the chance of none first, then take it from 100%.

Traps to avoid

  • 1/11 for a total, because there are eleven totals from 2 to 12. They are not equally likely — count ordered pairs.
  • Adding chances for “at least one”: 4 × 1/6 for four rolls. That counts overlaps more than once and can pass 100%.
  • Counting a card that fits both parts twice: 17 hearts-or-kings instead of 16.
  • Forgetting that a drawn ball is not put back: (5/8)² instead of 5/8 × 4/7.
  • Ignoring what you were told, and answering with the chance over all 36 pairs instead of only the pairs that fit.

How to practise

  • Use Kinds of question to drill one kind that trips you — “At least one” or “Given that…”, say — on its own, then switch the rest back on.
  • Set Answer by to “Typing the answer”: you give a percentage, and within 0.5 counts. Turning 5/14 into about 35.7% is a skill of its own.
  • Learn the two-dice table by heart: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 ways for totals 2 to 12.
  • After a miss, read the working: it shows the count and the total it was divided by, so you can see which of the two you got wrong.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More probability

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