How Many Rolls to Get Every Number on a Die?
Roll a die until you have seen every number from 1 to 6. How many rolls does it take? Most people guess about 6 to 10. The answer is 14.7 on average, and quite often more than 20. The same maths explains why a random picker that can repeat names takes ages to get round a whole class, and why the last sticker in an album is so hard to find.
The Answer for a Die
The trick is to split the job into steps. The first roll always shows a new number. After that, each new number is harder to find, because more of the six faces are ones you already have:
| New number needed | Average rolls to get it |
|---|---|
| 1st (any number) | 1 |
| 2nd (5 of 6 are new) | 1.2 |
| 3rd (4 of 6 are new) | 1.5 |
| 4th (3 of 6 are new) | 2 |
| 5th (2 of 6 are new) | 3 |
| 6th (only 1 is new) | 6 |
| All six | 14.7 |
Add up the steps and you get 14.7 rolls. About 40% of that is the wait for the very last number: once you have five, only 1 face in 6 helps, so you need 6 rolls on average for that one alone.
How Lucky Can You Get?
The average hides a wide spread. Here are the exact chances of having seen all six numbers within a given number of rolls:
| Seen all six numbers within | Chance |
|---|---|
| 6 rolls (no repeats at all) | 1.5% |
| 10 rolls | 27% |
| 13 rolls | just over 50% |
| 20 rolls | 85% |
| 23 rolls | 90% |
So in one game in ten you will still be missing a number after 23 rolls. A stubborn missing number is not a sign that the die is broken, and it is not "due" either; every roll still has the same 1 in 6 chance. Thinking otherwise is the gambler’s fallacy. Try it yourself with our Dice Roller.
More Choices, Much Longer Waits
Mathematicians call this the coupon collector’s problem, after shoppers collecting a full set of coupons or cards. With more choices, the wait grows faster than the number of choices:
| Choices | Average draws to see them all |
|---|---|
| 2 (a coin) | 3 |
| 6 (a die) | 14.7 |
| 10 | 29.3 |
| 12 | 37.2 |
| 20 | 72 |
| 25 (a class) | 95.4 |
| 30 (a class) | 119.8 |
A rough rule: with n choices, expect about n times the natural log of n, plus another 0.58 times n. For 30 choices that is close to 120 draws, four times as many as there are choices.
Why This Matters for Picking Names
Imagine a teacher who picks a random student 30 times from a class of 30, and the same name can come up again. On average about 11 students are never picked at all, while others are picked twice or three times. To be sure everyone is picked once, the teacher would need about 120 picks.
That is why our Random Name Picker can remove each name after it is picked, and our Random Number Generator has a no-repeats option. Every name still has an equal chance on each draw, but nobody is left out and nobody is called twice before everyone has had a turn. See our guide to picking a random student fairly for more.
Stickers, Toys and Collections
The same problem appears whenever you collect a set at random: stickers in an album, toys in cereal boxes or cards in packs. If an album has 600 stickers and each one is equally likely, you need about 4,200 stickers on average to fill it, about seven times the size of the album, and most of the cost goes on the last few. Swapping doubles with friends is the easiest way to cut that down. (Real collections are often worse, because some items are printed less often.)
A Close Cousin: The Birthday Paradox
The coupon collector asks how long until every option has come up. The birthday paradox asks the opposite question: how long until any option comes up twice. That happens surprisingly fast. With a die, the chance of a repeat is already 44% within 3 rolls and 72% within 4, while seeing all six takes 14.7 on average.
FAQ
How many rolls does it take to get every number on a die?
On average 14.7 rolls. About half of all tries take 13 rolls or fewer, and about 1 in 10 still have a number missing after 23 rolls.
What is the coupon collector’s problem?
It asks how many random draws you need to collect every item in a set when each draw is equally likely to be any item. With n items, the average is n times (1 + 1/2 + 1/3 + ... + 1/n).
What are the odds of rolling 1 to 6 in six rolls?
About 1.5%, or 720 in 46,656. Every roll would have to show a number you have not seen yet.
How many picks until every student is chosen?
With a class of 30 and names that can repeat, about 120 picks on average. Removing names after they are picked gets everyone in 30 picks.
Why is the last number so hard to get?
Once you have all but one, only one face in six is new, so you wait 6 rolls on average for it, about 40% of the total.