Why small probabilities are hard to feel
People are reasonably good at judging chances such as one in two or one in six. Beyond about one in a thousand, most of us stop having a real feel for the number. “One in a million” and “one in a billion” both sound like “almost never”, even though one is a thousand times more likely than the other.
Comparisons help, but only if they are built from things we can count or imagine, and only if the arithmetic behind them is right. This article builds a few yardsticks and then looks at the limits of using them.
A yardstick from coin flips
A fair coin lands heads with probability one half. The chance of getting heads on every one of several flips in a row is found by multiplying, which is the multiplication principle at work:
- 10 heads in a row: 1 in 1,024
- 20 heads in a row: 1 in 1,048,576
- 23 heads in a row: 1 in 8,388,608
- 24 heads in a row: 1 in 16,777,216
For a hypothetical game where you pick 6 numbers from 49, a single ticket has a jackpot probability of 1 in 13,983,816. That sits between 23 and 24 heads in a row. If you can imagine flipping a coin two dozen times and seeing nothing but heads, you have a rough feel for the size of that number. Nobody would expect to see it happen on purpose in a single attempt.
A yardstick from dice
Rolling a particular number on one die is 1 in 6. All six dice showing a six at once is 6 × 6 × 6 × 6 × 6 × 6 = 46,656, or 1 in 46,656. Ten dice all showing a six is 6 multiplied by itself ten times, which is 60,466,176, so 1 in about 60 million. The 6-from-49 jackpot is much rarer than six sixes and much more common than ten sixes.
A yardstick from card hands
In poker, being dealt a royal flush in five cards has a probability of 4 in 2,598,960, which is 1 in 649,740. Most people never see one in a lifetime of casual play. Even so, the 6-from-49 jackpot is roughly 21 to 22 times rarer than that. The full table of hand frequencies is in hand rankings odds, and the deck that produces it is described under standard deck.
A yardstick from crowds
Another way to see a large number is to turn it into people. Suppose a stadium holds 70,000 people, a made-up round figure. Dividing 13,983,816 by 70,000 gives about 200. So a jackpot probability of 1 in 13,983,816 is like picking the single winning person out of roughly 200 full stadiums.
What counts as an everyday risk
Official statistics on accidents, illness, weather and similar events are often quoted next to lottery odds. Treat such comparisons with caution:
- The figures change by country, year, age group and activity, so a number quoted without a source and a date may not apply to you.
- A risk figure may be per year, per lifetime or per event, and mixing those units gives meaningless comparisons.
- Many everyday risks are not spread evenly across the population. An average over everyone can hide large differences between groups.
For these reasons this article uses constructed examples with exactly known probabilities rather than quoting accident or weather statistics. If you want to compare a real-world risk, look up the original source and check what exactly was counted.
Why the comparison is not the whole story
A comparison tells you how rare something is, but it leaves out three things.
The size of the stakes. An everyday risk is usually weighed against a benefit, such as getting to work. A lottery ticket costs a small amount and has a tiny chance of a very large prize. Whether that trade is sensible depends on the average return per ticket, not only on how rare the jackpot is. That calculation is explained in expected value, and the long-run result is typically a loss, as in any game with negative expectation.
Repetition. A tiny chance per ticket becomes a little less tiny over many tickets, but only a little. Buying one ticket every week for a year gives 52 tickets, which in the 6-from-49 illustration is 52 in 13,983,816, about 1 in 269,000. That is a better chance than a single royal flush hand, yet still a very small one. The law of large numbers describes long-run averages across many plays, not a guarantee that a rare event will arrive.
Someone does win. With very large numbers of tickets sold, a jackpot is sometimes won, which is not a contradiction. A rare event for each person can still happen to someone when many people take part. That is why seeing a winner on the news says little about your own chances.
Putting it to use
The best use of these comparisons is as a reality check before you spend money. If a tiny probability feels exciting, try restating it in coin flips: roughly 24 heads in a row. If you decide to play, keep the cost to an amount you would be comfortable spending on entertainment, and avoid the temptation of chasing losses after a run of unsuccessful tickets.
Gambling can be harmful, and no pattern of play changes the odds. If it is causing you worry or financial stress, a local support service can help.