the ledger · draw · EP-061

Price the Bet (Expected Value by Hand, on Three Everyday Bets)

Every bet has a second price underneath.

General education, not financial, insurance, or gambling advice — the worked figures are round toy examples, not any real product's odds. Read the odds a lottery publishes for the prize you are pricing — on the ticket or in its rules — before you buy; the Federal Trade Commission publishes plain consumer guidance on extended warranties and service contracts; and for consumer financial products the Consumer Financial Protection Bureau is the body to consult. For your own coverage, ask your insurer or a licensed agent. Nothing on this page is sponsored.

Study V — The Ledger.

the draw

What this draw covers.

Every bet you are offered has a price on the tag and a second price underneath it. This draw teaches the operation that finds it, by hand, in round numbers: chance times payoff, minus the price — worked on the lottery ticket, the extended warranty, and the school raffle. Then the two turns that make it the Ledger's tool and not a math lesson: why the same arithmetic waves off the warranty and buys the fire policy, and why a good bet lost is still a good bet. Three lines on paper before you answer the cashier.

What follows is the draw as spoken, line by line, with the gold line that sat under each.

On this page: the draw, in order · the rep · sources

as spoken

The draw, in order.

  1. Every bet has a second price underneath.

    You place bets all day and never price one. The ticket at the counter, the warranty the cashier is trained to offer, the raffle for the school roof, the policy you renew without reading. Each has a price on the tag and a second price underneath it, and almost nobody looks under.

  2. One operation. Three lines.

    On Monday this well said fear is a bad actuary, and the cure is to count. Tonight, the counting. One operation, three lines on paper, and you can price any bet you are offered for the rest of your life.

  3. Chance times payoff, minus the price.

    Here is the operation. Take the chance of winning, written as a fraction. Multiply it by what you win. That number is what the bet is actually worth — what it would pay you, on average, if you could make it over and over, without end. Then subtract what they are charging. What is left is the true price of the bet. Positive, the bet pays you. Negative, you pay the bet.

  4. The chance is usually printed. Read it.

    The hard part is never the multiplying. It is the first line — the chance. Where does it come from? From counting, the way Monday taught: how often does this happen, to people like me. And often it is simply printed. Every lottery publishes its odds — on the ticket or in its rules. Read the odds for the prize you are pricing.

  5. Worth one dollar. Sold for two.

    So price the ticket, in round numbers kept simple on purpose. A two-dollar ticket. Say the odds of the big prize are one in ten million, and the prize is ten million dollars. Ten million, times one ten-millionth, is one dollar. That is what the ticket is worth. They are charging two. The true price is minus one dollar — half of every dollar gone before the drawing is held.

  6. The number is the price of the dream.

    Real tickets carry smaller prizes too, and real odds differ from my round ones — so read yours. But the shape seldom changes: a lottery is a pool that keeps part of what goes in, and the arithmetic will usually show you that part. The number is not an insult. It is the price of the dream, stated honestly.

  7. Worth thirty. Sold for two hundred.

    Now the counter. A two-hundred-dollar warranty on a thousand-dollar appliance. Suppose one in ten fails inside the window and the average repair is three hundred dollars. One tenth of three hundred is thirty. Thirty dollars is what the warranty is worth. They want two hundred. True price: minus one hundred seventy — the worst everyday bet in the building.

  8. Worth a dollar and a quarter. Sold for five.

    Now the raffle, because it teaches something the others cannot. Two hundred tickets at five dollars each, for a basket worth two hundred fifty. Your chance is one in two hundred. Two hundred fifty divided by two hundred is a dollar and a quarter. You paid five. True price: minus three dollars and seventy-five cents.

  9. A bad bet can be a good purchase — on purpose.

    And here is the lesson. That is a terrible bet and a perfectly good purchase — as long as you know which one you are making. Three dollars and seventy-five cents of your five is a gift to the school roof. The arithmetic does not tell you not to buy the ticket. It tells you what you are actually buying, so you can buy it on purpose.

  10. Minus three hundred. And you buy it anyway.

    Now the first turn, and it is the one that separates a Ledger from a calculator. The sign matters — but so does the size. Price the roof over your head. Say three houses in a thousand burn in a year, at three hundred thousand dollars each. Three thousandths of three hundred thousand is nine hundred dollars — the expected loss. The policy costs twelve hundred. True price: minus three hundred a year.

  11. The average exists only for bets you survive.

    By the number alone, that is a bet against you, like the warranty. And you buy it anyway — because the operation quietly assumed you could play over and over, and you cannot lose your house over and over. You can lose it once. The average only exists for bets you survive. So the rule has two halves: price the bet, then ask whether you can afford the losing side. Small loss, trust the number. Ruin, buy the floor — if you can afford its price. That is why the same arithmetic waves off the warranty and buys the fire policy.

  12. The number says yes. The size says no.

    It runs the other way too. Someone offers you a coin: heads you win three dollars, tails you pay one. Half of three is a dollar fifty. The price here is not on a tag — it is the half chance of paying one: fifty cents. The difference is a dollar in your favor, on average, every flip. Take that bet all day. Now the same coin, but heads triples your savings and tails takes all of it. The number still says yes. The size says no.

  13. A good bet lost is still a good bet.

    And the second turn, which is the one people cannot hold. A good bet lost is still a good bet. Take the three-dollar coin, lose the flip, and you were still right. Buy the ticket, win, and you were still wrong — you were paid, but you were not right. Judge the decision by the price you paid for it, never by the way the coin fell.

  14. The world grades outcomes. The coin doesn't care.

    This is the whole discipline of the Ledger, and it is hard because the world grades you by outcomes and the coin does not care. The gambler who won remembers the win. The counter who lost writes down the price and makes the same bet tomorrow.

the rep

Three lines before you answer.

So here is your work this week, and it costs a pencil. The next bet you are offered — the ticket, the warranty, the raffle — write the three lines before you answer. The chance, as a fraction, from the ticket, its rules, or the count. Times the payoff. Minus the price. Then ask the second question, out loud: can I afford the losing side? Buy or decline on that.

The well is deep. Bring a bucket.

drawn from

Where it comes from, and where it stops.

Drawn from: plain expected-value arithmetic, worked through in the episode and kept as round-number toy examples — the ticket ($2 for a 1-in-10,000,000 chance at $10,000,000 = $1.00 expected payout, −$1.00), the warranty ($200 against a 1-in-10 failure at $300 = $30, −$170, the same figures as EP-047), the raffle (200 tickets at $5 for a $250 prize = $1.25, −$3.75), the fire policy (3-in-1,000 at $300,000 = $900 expected loss against a $1,200 premium, −$300, the same figures as EP-047), and the coin (½ × $3 − ½ × $1 = +$1.00).

General education, not financial, insurance, or gambling advice — the worked figures are round toy examples, not any real product's odds. Read the odds a lottery publishes for the prize you are pricing — on the ticket or in its rules — before you buy; the Federal Trade Commission publishes plain consumer guidance on extended warranties and service contracts; and for consumer financial products the Consumer Financial Protection Bureau is the body to consult. For your own coverage, ask your insurer or a licensed agent.

read alongside it

From the shelf.

As an Amazon Associate I earn from qualifying purchases. The commission costs you nothing and bought none of these; the whole shelf is on the reading page, and how that works is on the about page.

  1. Against the Gods — Peter L. Bernstein

  2. How Not to Be Wrong — Jordan Ellenberg

keep drawing

More from The Ledger.

drink deep

The rest of the bucket.

The Codex of Nine is the free twelve-page primer to the well every draw comes from: one page per domain — the question it answers, three truths already drawn, one thing to do this week.

After that, one note per episode — the video and its sources — twice a week and nothing else. Unsubscribe in one click.