Published · 2026-10-28 · Last reviewed · 2026-10-28 · 5 min read

One probability, five wordings

The chance of a Lotto 6/45 first prize is 1 in 8,145,060. You can say it in other ways: "0.0000123%", "about 1.2 tickets in 10 million", "buying one ticket a week, once in about 156,000 years on average", "buying 52 tickets in a year gives a 0.00064% chance", "the chance of missing is 99.9999877%". All are the same number, yet they feel very different. This page rewrites one probability five ways, shows how the wording can change judgement, and collects a few habits for reading numbers.

One probability, five wordings

Five wordings of the Lotto 6/45 first-prize probability 1/8,145,060 (exact conversions)
Style	Sentence	How it tends to feel
Ratio	1 in 8,145,060	a big number comes into view
Percent	0.0000123%	a string of zeros that looks like nothing
Natural frequency	about 1.2 tickets in 10,000,000	turns into a story at a size people can imagine
My own time scale	one ticket a week: once in about 156,636 years on average	invites comparison with a lifetime
Chance in a year	52 tickets: 0.00064% to win at least once	takes the shape of "what about this year?"

All five lines are one value. But the middle ones touch the scale of human experience, so "how rare" is felt more sharply. That the same number worded differently can affect judgement has been studied for a long time. Every number here is a conversion of 1/8,145,060.

Making the scale imaginable

There are many ways to turn 8,145,060 into something one can feel. Counting one ticket per second would take 94.3 days to count 8,145,060 tickets. Assuming a stadium holds 50,000 people, 8,145,060 people fill 163 stadiums. These comparisons are for explanation, and none is "the accurate wording". Which comparison is used changes how the listener feels, so both speaker and reader should know it is a choice. The problem of scale is discussed further in Why big numbers blur our judgement.

Change the number

Change the "1 in …" value in the tool below. The default is the Lotto 6/45 first prize; you can enter the first-prize odds of another lottery or the probability of an everyday event. The same value comes back in five wordings.

The same exercise for each prize tier

Rewriting the probabilities of all five tiers the same way shows which wording suits which size. Fifth prize, the most likely, reads easily as a percent, the middle tiers as "out of 1,000", and the very rare ones as "out of a million".

Lotto 6/45 tier probabilities in several wordings (exact conversions)
Tier	Ratio	Percent	Per 1,000 tickets	Per million tickets
5th	1 in 44.6	2.24%	22.4	22,440
4th	1 in 732.8	0.136%	1.36	1,365
3rd	1 in 35,724	0.0028%	0.028	28
2nd	1 in 1,357,510	0.0000737%	0.0007	0.74
1st	1 in 8,145,060	0.0000123%	0.0001	0.12

"22 in 1,000 tickets" lands at once for fifth prize, but for first prize "0.0001 in 1,000 tickets" loses its meaning. The reason to change wording is to let a person imagine the number at a scale that fits. It is better to say that different scales suit different sizes of number than that one wording is better.

Relative risk and absolute risk

A typical case where wording sways judgement is "times as much" versus "out of how many". Take an imaginary drug (invented numbers). Without the drug 2 in 1,000 people get a disease, and with it 1 in 1,000. That can be said as "the risk is cut by 50%" or as "if 1,000 people take the drug, one fewer gets ill (an absolute reduction of 0.1 percentage points)". The first sounds big and the second small, but they are the same fact. The relative figure (50%) alone makes the benefit feel large; adding the absolute figure (1 in 1,000) shows its size. Giving both reduces misunderstanding.

In lottery adverts or statistics too, a relative phrase such as "twice as many winners" gives no size unless it says from what to what. If a round had 16 first-prize winners, twice the usual, one ticket's chance of winning was not changed. When talking about the size of a change it helps to look at "what, from what to what" together.

The same thing as a gain or as a loss

Tversky and Kahneman (1981) reported that when the same problem is worded as a gain or as a loss, people's choices shift in a predictable way. According to a summary of the abstract, reversals of preference were shown both in choices about money and in questions about the loss of human lives. I could not open the body of the paper and quote no specific percentages.

Here is the structure with an everyday example (invented numbers). "The five-year survival rate of this operation is 90%" and "10% of patients die within five years" state one fact, yet people's reactions can differ. With a lottery, if "top prize up to KRW 4 billion" (a gain wording) is placed beside "chance 1 in 8,145,060" (a probability wording), the first sentence is the one that catches the eye. We cannot know which wording will be noticed first, but remembering that every wording comes from one fact makes judgement less easy to sway.

Switching to natural frequencies

Gigerenzer and Hoffrage (1995) reported that when probabilities are presented as frequencies such as "out of a hundred people", people do much better at Bayesian reasoning (such as how likely a positive test is to mean disease); I checked a summary. What a positive test really means uses the same method. Instead of "prevalence 1%, sensitivity 90%, false-positive rate 9%" write "10 of 1,000 people have the disease and 9 of them test positive; of the 990 without it about 89 test positive", and you see at once that among the 98 positives only 9 have the disease (about 9%). The same calculation, only the wording changed.

Habits for reading numbers

Check the denominator. "Down 10%": 10% of what? If "mortality doubled", what was it before? A rate of change without a base says nothing about size.
Convert to natural frequencies. Turning a probability into "how many out of 1,000" shows the scale. It helps most when the percentage is very small or very large.
Convert to my time scale. Knowing how long "once" takes to come (here about 156,000 years) helps judge whether it will happen to me.
Look at the opposite wording. "90% success" is the same as "10% failure". Which one is printed may be intentional.
Turn it into money. Prize times probability gives a ticket's expected value. For the Korean lottery, KRW 1,000 returns KRW 500 on average (Why do people buy lottery tickets with negative expected value?).

Questions that come up

"Which wording is the most honest?" Honesty lies in the completeness of the information, not in the wording. Showing two or three wordings together reduces misunderstanding more than choosing one.

"So do lottery adverts deceive?" This page evaluates no particular advert or operator. It only shows that how the same fact is put can change the impression.

What the comparisons assume

How much wording affects judgement varies greatly with the situation and the person. Both papers were checked only through abstract-level summaries, and this page quotes no figures or effect sizes from their text. The scale comparisons (94.3 days, 163 stadiums) are only assumptions to help understanding. The operation survival example is an explanatory sentence, not real data.

One probability, five wordings