Warranties, subscriptions, big purchases, pawn loans, a degree. Pick what you are deciding, start from a common example or your own numbers, and see the one figure that settles it.
Across: the chance. Up: the payout. The black line joins every single outcome that would pay back exactly the price (chance × payout = price). A one-outcome deal above the line comes out ahead on average. With several outcomes, their payback adds up, which the bar above shows.
chance × payout = average payback
Multiply to gather the deal into one number: what it returns on average each time. Several outcomes? Add their products.
price ÷ payout = break-even chance
Divide to split it back apart: how likely the outcome would have to be for the price to be fair.
times × (price − payback) = expected loss
Repeat a losing deal and the loss grows with every go. That is why an hour of slots costs far more than one spin.
Preset numbers are illustrative, not quotes, except the Powerball prize odds (the published odds for each tier; the jackpot varies) and the payday example, whose $15 per $100 fee is the typical figure the US Consumer Financial Protection Bureau cites. Real slot machines do not publish their odds; the slot presets use a typical 92% return. Nothing you type leaves this page; your own numbers are remembered in this browser only.