Why tiny numbers are hard
Slot machines and other games of chance have a built-in house edge, and no strategy, system or timing changes the odds. Most players lose money over time, and nobody should gamble money they cannot afford to lose. This article uses ordinary comparisons to build a feel for very small chances. The figures are general or hypothetical, and none is a prediction about any game.
Human intuition is tuned for everyday quantities. We can picture a handful of apples, perhaps a crowd of a thousand. Past that, “a lot” and “a huge lot” blur together, and so do “unlikely” and “very, very unlikely”. Rare events are where this blurring does the most damage to our judgment.
What one in a million looks like
One in a million is 1/1,000,000, which is 0.0001%, or 0.000001 as a decimal. Here are some ways to feel the size.
Seconds. A million seconds is about 11.6 days (1,000,000 ÷ 86,400 is about 11.57). A billion seconds is around 31.7 years. If one particular second out of a million were secretly marked, picking it blindly from the last eleven and a half days would be a one-in-a-million shot.
Coin flips. Twenty heads in a row on a fair coin has a probability of 1 in 1,048,576, so very close to one in a million. Imagine flipping a coin twenty times and having every single flip come up heads.
Dice. Rolling six dice and getting a six on every one has a probability of 1 in 46,656, since 6 multiplied by itself six times is 46,656. That is roughly twenty-one times more likely than one in a million. You would need around nine dice to be in one-in-ten-million territory, since 6^9 is 10,077,696.
These are all instances of compound probability, where multiplication quickly shrinks a chance.
Lower is not always “close to zero”
A common habit is to treat any very small chance as zero. It is not zero, and over large populations or many repeats, rare events do occur.
Suppose a hypothetical event has a one-in-a-million chance on any given day, independent from day to day. Over 365 days the chance that it happens at least once is roughly 365 in a million, about 0.0365%. That is still small, but it is 365 times larger than the single-day figure.
Meanwhile, if a million people each face a one-in-a-million chance, the expected number of occurrences is one. It would not be shocking if it happened to someone, but it would be very surprising for it to happen to you in particular. This difference between “to someone” and “to me” is why rare events make the news.
Comparing small chances to each other
Tiny numbers can differ by factors that people overlook. One in 10,000 is 0.01%. One in 1,000,000 is 0.0001%. The first is a hundred times more likely. If a game or contest quotes a chance in the millions, it can be worth calculating how many repeats it would take to reach even a modest cumulative probability.
For a hypothetical 1 in 10,000,000 chance per attempt, 1,000 attempts give a chance of about 1,000 ÷ 10,000,000 = 0.01%. This approximation is accurate for small chances. And 0.01% is still tiny. The expense of making 1,000 attempts, however, could be considerable.
Why we misjudge small probabilities
Several mental habits distort our sense of rare events.
- Vividness. Dramatic outcomes are easy to imagine and recall, so they feel more likely than the numbers suggest. This is one of the well-known cognitive biases.
- Selective memory. We hear about the rare winner, not the very many who did not win. That makes the event seem common. Selective memory of wins reinforces the impression.
- Rounding up. Saying “it could happen” slides into “it might happen to me”.
- Percentages that hide the scale. “0.0001%” has so many zeros that we skim over it. Using “1 in N” statements or counting people can help.
The best antidote is to translate the chance into a crowd. A one-in-a-million chance is about one person among 20 full stadiums of 50,000 people each.
Long runs and short sessions
For repeating events, the law of large numbers tells us that frequencies approach the underlying probability only after very many trials. For a chance of one in a million, “very many” means millions. A person who plays a few hundred or a few thousand rounds of anything with such a small chance will almost certainly see nothing, and that tells them nothing about whether the chance is fair or whether a result is “due”. Believing an outcome is overdue is the gamblers fallacy.
Large prizes with tiny odds are also the reason expected value can be negative even when a headline number looks large. A big payoff multiplied by a very small chance may still add up to less than the amount staked.
A few practical habits
- Convert small chances to “1 in N” and then to a crowd you can picture.
- Compare the chance with another familiar rare event of known size, such as twenty heads in a row.
- Ask whether a chance is per attempt, per day or per lifetime.
- Be wary of promises that downplay the scale of a long shot, as in unrealistic promises of an easy big win.
Keeping it in proportion
Seeing how small a tiny probability really is can be sobering. If you gamble, treat it as paid entertainment, set limits before you start and never chase losses. If gambling stops being fun, consider a break, and know that a national problem-gambling helpline or local health service can provide confidential support.