When a die is thrown, there is no knowing what will come up. Over thousands of throws, the share of aces becomes almost predictable.
The question
A die lands on the ace one time in six: is that true over six throws?
What you do Throw 6,000 dice one by one and watch the share of aces narrow. Then throw whole series, as many as you like: each one leaves a dot.
Without JavaScript, here are the animation’s data.
| Throws | Aces expected on average | In 95% of series, number of aces | That is, a share of aces of |
|---|---|---|---|
| 6 | 1 | 0 to 3 | 0 to 50% |
| 60 | 10 | 5 to 16 | 8.3 to 26.7% |
| 600 | 100 | 82 to 118 | 13.7 to 19.7% |
| 6,000 | 1,000 | 944 to 1,057 | 15.7 to 17.6% |
Over 6 throws, there is no ace about one time in three (33%). Over 6,000 throws, the number of aces is more than 60 away from 1,000 in only 3.6% of series: “very rarely”, as the article says.
These figures are calculated exactly (binomial distribution), for a fair die.
The takeaway A small number of throws can give almost anything. The more you throw, the more tightly the share of aces gathers around one in six: this is the law of large numbers.
Sources 1
Bernoulli, Jacques, Ars Conjectandi, Bâle, 1713, public domain ; partie IV traduite en anglais : Sheynin, Oscar, On the Law of Large Numbers, Berlin, 2005, part IV, chapter 4.
even the most stupid person, all by himself and without any preliminary instruction, being guided by some natural instinct […] feels sure that the more such observations are taken into account, the less is the danger of straying from the goal
Public domain and open access open the source · checked on 24 October 2026