What is expected value?

Expected value is the probability-weighted average of every possible outcome of a bet · what you would get back on average if you could repeat it many times. You compute it by multiplying each result by its chance and adding the products together.

Why do rare jackpots and risk of ruin matter so little, or so much?

A rare outcome contributes only its result times its small probability, so a 10% jackpot adds a tenth of its value to the average · less than people expect. A risk of ruin works the same way but downward: even a small chance of losing everything pulls the expected value down and, in real life, can end the game entirely.

How does difficulty work?

Easy is a two-outcome gain-vs-loss gamble. Medium adds a third outcome and a rare jackpot. Hard is a four-outcome bet with both a jackpot and a risk of ruin · the fat-tailed gambles where the arithmetic matters most.

Can I print it as a worksheet?

Yes. Expected Value supports the printable worksheet generator at /worksheets · each problem prints the table of chances and results with lettered choices and an answer key, ready for a maths or finance classroom.

Does it work offline?

Yes · Expected Value is part of the PlayMemorize Progressive Web App. Install once and play anywhere, even with no connection.

Expected Value

Estime o retorno ponderado pela probabilidade de uma aposta · avalie ganhos, perdas, jackpots e o raro risco de ruína

FAQ

About Expected Value

Expected Value is a fast decision-making trainer built on the single most useful idea in probability: the expected value of a bet is the average amount you would walk away with if you could play it over and over. Each round shows a description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',00 stake and a small table of outcomes · each outcome has a chance and a dollar result · and your job is to estimate the probability-weighted average and pick the closest figure from four choices.

How expected value works. You multiply each result by its probability, then add everything up. A bet with a 50% chance of ending at $200, a 10% shot at a description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',,000 jackpot, a 30% chance of dropping to $50, and a 10% risk of losing it all is worth 0.5 × $200 + 0.1 × description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',,000 + 0.3 × $50 + 0.1 × $0 = $215 on average. That single number tells you whether the bet is worth taking · above your description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',00 stake it is favourable, below it is a losing proposition.

Why the tails matter. The whole skill is learning to weight rare outcomes correctly. A jackpot that only hits 10% of the time barely moves the average · and a 10% risk of ruin quietly drags it down. Most people anchor on the headline outcome or average the results as if every branch were equally likely; the game's distractors are built from exactly those mistakes, so you learn to spot them in yourself.

Three difficulty tiers. Easy is a clean two-outcome gamble · one gain, one loss · the calculation a beginner can do in their head. Medium adds a third branch and a rare jackpot. Hard is the full four-outcome bet with both a jackpot and a risk of ruin, the fat-tailed gambles where intuition fails and the arithmetic earns its keep.

Deterministic. Every round is generated from a seed, so the same difficulty always produces reproducible bets and the printable worksheet's answer key matches the live game byte-for-byte.

Part of the PlayMemorize family of brain-training games. Runs entirely in your browser, works offline as a Progressive Web App. Prefer to just crunch your own numbers? Use the Expected Value Calculator.

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FAQ

What is expected value?

Expected value is the probability-weighted average of every possible outcome of a bet · what you would get back on average if you could repeat it many times. You compute it by multiplying each result by its chance and adding the products together.
How do I calculate it quickly?

Multiply each result by its probability (as a decimal) and sum them. For example, a 50% chance of $200 plus a 10% chance of description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',,000 plus a 30% chance of $50 plus a 10% chance of $0 is description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',00 + description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',00 + description: 'Estimate the probability-weighted payoff of a bet · weigh upside, downside, jackpots and the rare risk of ruin.',5 + $0 = $215. Compare that to your stake to decide whether the bet is worth it.
Why do rare jackpots and risk of ruin matter so little, or so much?

A rare outcome contributes only its result times its small probability, so a 10% jackpot adds a tenth of its value to the average · less than people expect. A risk of ruin works the same way but downward: even a small chance of losing everything pulls the expected value down and, in real life, can end the game entirely.
How does difficulty work?

Easy is a two-outcome gain-vs-loss gamble. Medium adds a third outcome and a rare jackpot. Hard is a four-outcome bet with both a jackpot and a risk of ruin · the fat-tailed gambles where the arithmetic matters most.
Can I print it as a worksheet?

Yes. Expected Value supports the printable worksheet generator at /worksheets · each problem prints the table of chances and results with lettered choices and an answer key, ready for a maths or finance classroom.
Does it work offline?

Yes · Expected Value is part of the PlayMemorize Progressive Web App. Install once and play anywhere, even with no connection.