Thinking in Odds

Six short lessons on chance: what "likely" means, why a coin has no memory, how to weigh a rare thing, what a fair price is, what a bingo hall keeps, and what the banker keeps at a festival dice game.

Spark & Whetstone · sparkwhetstone.org/learn/thinking-in-odds · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. What "likely" means

When we say something has a one-in-six chance, we are making a claim about the long run. Roll a fair die six hundred times and you should see a six roughly a hundred times — not exactly, but close, and closer the longer you go.

That is all a probability is: a number between 0 and 1 for how often something would happen if you could repeat the situation many times. 0 means never, 1 means always, and one half means half the time.

Short runs are lumpy. Ten rolls can easily show three sixes, or none. The pattern only settles down with many tries — which is why a single surprising result tells you very little.

Question. You roll a fair die 12 times and get no sixes. What is the best way to read that?

The die is probably unfair.
Twelve rolls is a short run; no sixes is unusual but quite possible.
A six is now due on the next roll.

Say it back. In your own words: what does "a one-in-six chance" actually promise, and what doesn't it promise?

________________________________________________

Lesson 2. A coin has no memory

Suppose a fair coin comes up heads five times in a row. Many people feel tails is now "due". It isn't. The coin does not know what it did before; each toss is still one half.

This feeling has a name — the gambler's fallacy — and it is ordinary, not foolish. Our sense of pattern expects runs to even out quickly. They do even out, but only by being swamped by many later tosses, never by the coin correcting itself.

The useful habit is to ask: are these events independent? For coins and dice, yes. For things with memory — a deck you are dealing from, a queue that is shrinking — the past really does change the odds.

Question. A fair coin has landed heads five times in a row. What is the chance the next toss is heads?

Less than one half
Exactly one half
More than one half

Say it back. Say it back: when does the past change the odds, and when doesn't it? Give one example of each.

________________________________________________

Lesson 3. Weighing a rare thing

Picture 1,000 emails. 10 of them are scams. A filter catches 9 of those 10 scams — but it also wrongly flags 50 of the 990 honest emails.

So the filter flags 59 emails in all: 9 scams and 50 honest ones. If an email is flagged, the chance it is really a scam is 9 in 59 — about 15%. Most flagged emails are honest, even though the filter is "90% accurate" at catching scams.

This is the idea of a base rate. When the thing you are looking for is rare, even a good test turns up mostly false alarms. Counting in whole numbers — out of a thousand — makes this far easier to see than percentages.

Question. In the example, an email has just been flagged. Roughly how likely is it to be a scam?

About 90%
About 50%
About 15%

Say it back. In your own words: why can a good test still be wrong most of the time when the thing is rare?

________________________________________________

Lesson 4. A fair price

A library sells 500 raffle tickets at $2 each for a $500 prize. What is one ticket worth, on average?

Across all 500 tickets, the library pays out $500, so each ticket carries $1 of prize on average. You pay $2 for $1 of expected prize — on average you give $1 to the library. That is the expected value: the average outcome over many repeats.

That is fine for a fundraiser; you are paying for a good cause and a little fun. Expected value is not a rule about what to do. It is a way of seeing clearly what a price buys, so the choice is yours with your eyes open.

Question. A game costs $3 to play. One time in four you win $8; otherwise nothing. On average, per play, you…

gain $1
break even
lose $1

Say it back. Say it back: what does expected value tell you about a price — and what does it leave to you?

________________________________________________

Lesson 5. The bingo hall

A hall sells 100 cards at $2 and pays $150 to whoever finishes the first line. It takes in $200 and pays out $150, so it keeps $50 a game. Each card, on average, carries $1.50 of prize for its $2 price: the players get back 75 cents of every dollar they pay.

Nobody in the room is being cheated. The cage is honest and every card has the same chance. The keeping is in the price, decided before a ball is drawn — which is why the numbers belong on the wall where everyone can read them.

Bingo apps keep more quietly. Coins you buy turn into tickets you spend, so the price gets hard to see. A banner shouts SO CLOSE! when one square is missing, though "one away" on a card is ordinary. A daily bonus grows only if you come back. None of these changes the odds; they change how often you play.

Question. A hall sells 50 cards at $4 and pays a $120 prize. What share of what players pay comes back to them?

About 40%
About 60%
About 80%

Say it back. In your own words: where does a hall make its money, and what do the extra parts of a bingo app add?

________________________________________________

Lesson 6. The festival dice

At Lunar New Year many families play a three-dice game, bầu cua cá cọp — gourd, crab, fish, tiger — with six pictures instead of numbers. You put a coin on a picture and three dice are rolled. If your picture shows on one die, you win your coin back plus one more; on two dice, plus two; on three dice, plus three. If it shows on none, the banker takes your coin.

It feels close to fair. There are three dice and each has a one-in-six chance, so a picture turns up about half the time. But count every way the dice can land: 6 × 6 × 6 = 216. Your picture is missing from 125 of them, shows once in 75, twice in 15 and three times in just 1.

Bet $1 each time across all 216. You lose $125 and win back $75 + $30 + $3 = $108. That is $17 lost for every $216 staked, about 8 cents of every dollar, kept by the banker. The extra prize for doubles and triples never quite makes up for the times your picture misses. It is a family game, played for small coins, and the numbers belong on the table where everyone can see them.

Question. Over a long festival, players stake $200 in all on this game. About how much does the banker keep?

About $2
About $16
About $50

Say it back. In your own words: why does the festival dice game feel fair, and where does the banker's share come from?

________________________________________________

At the table: The two-dice table

Players: 2 or more, any ages. You'll need: Two dice, a pencil, paper.

Before rolling, each player guesses which total (2 to 12) will come up most often in 36 rolls.
Take turns rolling both dice 36 times in all. Tally each total.
Look at the tally together. Which total won? Most tables see 7 — there are six ways to make it, and only one way to make 2 or 12.
Talk it over: whose guess was closest, and why is the tally lumpy even though 7 is the favourite?

Answers

Twelve rolls is a short run; no sixes is unusual but quite possible. — A fair die shows no six in twelve rolls about 11% of the time. Surprising once is not evidence of much; it takes many rolls to judge a die.
Exactly one half — Still one half. Independent events have no memory. The streak was unusual, but it does not push the next toss either way.
About 15% — 9 flagged scams out of 59 flagged emails is about 15%. Because scams are rare, the false alarms outnumber them.
lose $1 — The average prize is $2 a play. You pay $3, so on average you lose $1 each time.
About 60% — The hall takes in $200 and pays $120, so players get back 60% — the hall keeps $80 a game.
About $16 — $200 × 17/216 ≈ $15.74, so about $16. The players get back about $184 of their $200.