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How much to bet

A bet with an edge, made again and again: how much of your bankroll to stake. The Kelly fraction, why over-betting loses, and what half-Kelly keeps.

  • Money
  • Untimed
  • Endless levels

How it plays

A bet pays b to 1 and wins with chance p. Its expected profit per unit staked is pb − q. If you make it many times, staking a fraction f of your bankroll each time, the bankroll grows fastest at the Kelly fraction f* = p − q/b.

Levels go from the edge, to the Kelly fraction, to choosing the fastest-growing stake among several, to over-betting — where a bet you are sure to win on average still shrinks your money — and to how much growth half-Kelly keeps.

This is mathematics about repeated bets with a known win chance. Real bets have an unknown one, and every question says so. Fractions are to 2 dp, within 0.01. The wrong options are the real slips: betting your win chance, the edge not divided by the odds, the loss forgotten.

Keys

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

What it trains

This makes reasoning about the size of repeated bets with known odds quicker and surer. It is not betting or investment advice and claims nothing wider.

Learn

How to get better at how much to bet: 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. Write down three numbers: p, the chance of winning; q = 1 − p, the chance of losing; and b, what a win pays per unit staked (“2 to 1” is b = 2).
  2. Work out the edge, pb − q: what one unit staked earns on average. If it is not positive, no stake is worth placing.
  3. For how much to bet, divide the edge by the odds: f* = (pb − q) ÷ b, the Kelly fraction of the bankroll.
  4. For the later questions, compare the stake asked about with f*: below it, at it, around twice it, or beyond.

Techniques and tricks

  • Expected profit of a stake

    Win p of the time and gain b per unit; lose q of the time and lose the unit. For a stake of 100: 100 × (pb − q).

    e.g. 2 to 1, winning 40%: 0.4 × 200 − 0.6 × 100 = 80 − 60 = 20.

  • Kelly is edge over odds

    f* = (pb − q) ÷ b, which is the same as p − q ÷ b. Use whichever is quicker for the numbers in front of you.

    e.g. 2 to 1 at 40%: 0.20 ÷ 2 = 0.10, or 0.4 − 0.6 ÷ 2 = 0.10.

  • Even money: twice p, less 1

    At 1 to 1, f* = p − q = 2p − 1. The edge and the Kelly fraction are the same number.

    e.g. Winning 55% at even money: f* = 0.10.

  • The break-even chance

    A bet at b to 1 is fair when p = 1 ÷ (b + 1). Above that there is an edge; below it, none.

    e.g. At 2 to 1 you need better than 1 in 3, 33.3%.

  • Twice Kelly grows nothing

    Growth per bet is roughly zero at about twice the Kelly fraction. Stakes well under that still grow the bankroll; stakes well over it shrink it, even though every bet has a positive average.

  • Half-Kelly keeps about three quarters

    Near its peak, growth is close to a parabola in the stake, so half the Kelly stake keeps about 1 − ¼ = ¾ of the growth, with half the swings. For every bet this game draws, it keeps between 73% and 79%.

As the levels climb

  • The kinds unlock one at a time: is there an edge at level 0; the Kelly fraction at level 1; which fraction grows fastest at level 3; over-betting at level 5; half-Kelly at level 7.
  • In “Which fraction grows fastest”, the options are multiples of f* such as a quarter, half, once, one and a half or twice it. Work out f* and choose it if it is there; if it is not, the nearest option usually wins, and two sitting either side of it come out close — the working shows each one’s growth, so check which won.
  • In over-betting, the stake asked about is between 1.5 and 3 times f*. Under twice f* it almost always grows; over twice, it shrinks.
  • After level 7 the bets themselves stop changing; further levels only add a little time. Use the extra time to type the answers instead of choosing them.

Traps to avoid

  • Betting your win chance: staking p of the bankroll because you win p of the time. Kelly is p less q ÷ b, so p is always too much, and at the short odds here often several times too much.
  • The edge not divided by the odds: pb − q given as the fraction. At 2 to 1 that is twice the Kelly fraction, right at the point where growth stops.
  • The even-money formula, p − q, used at other odds. It is only right at 1 to 1.
  • In the expected profit, the loss forgotten: p × b × 100 given as the profit. Always subtract what the losing bets cost.

How to practise

  • Use “Kinds of question” to drill the edge and the Kelly fraction until p − q ÷ b is automatic; every later kind starts from it.
  • Switch Answer by to “Typing the answer”: fractions are to 2 dp and count within 0.01, which makes you finish the division.
  • Remember what the questions say each time: the win chance is known exactly here. In real bets it is a guess, and the Kelly fraction of a wrong guess is a wrong bet.

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.

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