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Expected value

What a bet is worth on average: a paid spin, a fair price, a raffle ticket, an insurance policy, a wheel's house edge, and which of two bets is better.

  • Probability
  • Untimed
  • Endless levels

How it plays

Each question is a game of chance with known odds: pay to spin, price a game fairly, choose between two bets, value a raffle ticket, see a policy from the insurer's side, or find the edge a wheel keeps.

Expected value is each outcome times its chance, added up — the average result over many plays. New kinds unlock as you climb and the prizes, spinners and wheels grow without end. Gains are shown with a sign: + for a gain, − for a loss.

The wrong options are the real slips: forgetting the price you paid, taking the stake off only when you lose, counting a prize once when several are on offer, reading "pays 35 to 1" as 35 back. The games are invented; real bets do not tell you their chances. The money is in the currency you pick.

Keys

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

What it trains

This makes expected-value arithmetic of these kinds quicker and surer. It is not betting or financial advice and claims nothing wider.

Learn

How to get better at expected value: 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. Read whose side you are on and what is asked: your gain after paying, a fair price, the better of two bets, a ticket’s worth, the insurer’s profit, or a wheel’s edge.
  2. List each outcome with its chance and its money. On a spinner of equal sections, each section’s chance is 1 ÷ the number of sections.
  3. Multiply each money amount by its chance and add them up: that is the average result per play over many plays.
  4. Take off what you pay every time, if anything. Then check the sign: + is a gain, − a loss, and the options show both.

Techniques and tricks

  • Winning sections × prize ÷ sections

    A spinner with k winning sections out of n pays k/n of the prize on average. Take the stake off once — you pay it on every spin.

    e.g. ₹64 to spin, 4 of 8 sections win ₹160: 4/8 × 160 = 80, so +₹16 a spin

  • A fair price is the plain average

    When every section is equally likely, the expected win is just the average of the amounts on them. The fair price is that average.

    e.g. ₹33, ₹17, ₹0, ₹4, ₹11: 65 ÷ 5 = ₹13

  • Compare chance × prize

    For two bets, work out each one’s chance × prize and compare. A sure amount is its own expected value.

    e.g. A: 40% of ₹300 = ₹120. B: 25% of ₹500 = ₹125. B is worth more on average

  • A raffle is its prize pot

    Add every prize, divide by the tickets sold, and take off the ticket price. You do not need the chance of each prize.

    e.g. 1,000 tickets at ₹50, prizes worth ₹30,000 in all: ₹30 − ₹50 = −₹20 a ticket

  • Insurer’s side: premium minus expected claim

    The expected claim is the loss ÷ the “1 in N”. The insurer keeps the premium minus that; the buyer loses the same amount on average.

    e.g. ₹286 to cover ₹13,000 at 1 in 50: claim ₹260, insurer +₹26

  • The house edge from the payout

    A wheel with n pockets is fair if a one-pocket bet pays n − 1 to 1. Every step short of that costs 1/n of the stake. Here the payout is always 1 or 2 short.

    e.g. 20 pockets paying 17 to 1: 2 short, edge 2/20 = 10%, so −₹10.00 on ₹100

As the levels climb

  • You start with paid spinners and fair prices. Then the choice between two bets joins; then raffles and insurance; last the house edge of a wheel.
  • Between each new kind the numbers grow, without end: spinners of 12 and 20 sections and then more, bigger prizes, bigger raffles, bigger losses to insure, wheels with more pockets.
  • Bigger numbers do not change the method — they reward it. Simplify the fraction first (4/8 is a half), then multiply.
  • The time per question grows with the numbers. The game is untimed unless you turn on Time pressure.

Traps to avoid

  • Forgetting the price you paid: giving the expected winnings as the gain. A gain is after the stake.
  • Taking the stake off only on a win: k/n × (prize − stake). You pay to spin whether you win or not.
  • For a raffle, counting a prize once when several are on offer, or counting only the big prize. Add up the whole pot.
  • Reading “pays 35 to 1” as 35 back. You get your stake back plus 35 times it, so a win returns 36 times the stake.
  • For insurance, answering from the buyer’s side, or using the premium minus the whole loss. The insurer’s expected cost is the loss ÷ N, not the loss.

How to practise

  • Use Kinds of question to drill “The house edge” or “Insurance” alone if they lag, then switch the rest back on.
  • Set Answer by to “Typing the answer” to give the amount yourself, negative for a loss. Getting the sign right is half the skill.
  • Pick your own Currency, so the amounts feel like money you know; the sums are the same in every one.
  • Before reading the options, say out loud “gain” or “loss” and roughly how much. If the answer you then pick has the other sign, recheck the stake.

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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