Dice probabilities with multiple rolls
The previous post started with a lock pick during a session which felt too easy, and a blog post lamenting how doors have become push-button road bumps rather than parts of the world, and then dug into the question of how long lock picking should take.
In reviewing alternatives, I want what I pick to “feel” right, as in, for person of a given level of proficiency, does their probability of success match what I would expect.
To do that, I need to know the probability of success. And given that various schemes allow you to keep trying until someone disturbs you or you get bored, I need to know how the probability increases with each attempt.
This post is unavoidably a bit mathematical, so as well as explaining how to calculate the probabilities, I also provide pre-calculated tables so you can just look up the probabilities.
I also discuss how the probabilities change if you introduce a chance of instant failure, and the third edition concepts of “take 10” and “take 20”.
Single roll percentages
Since most of the schemes for lock-picking proficiency are based on d20 checks, and the ones which aren’t are based on multiples of 5% (at least at lower levels) which directly correlate to d20 checks, I will just concentrate on probabilities for d20 rolls.
Note that, although there are various DCs and bonuses or penalties, for the purposes of probability we can just collapse this to a roll against a target number. A DC15 check with a +7 bonus is effectively trying to roll 8 or higher, for example, as is a DC10 check with a +2 bonus.
The simplest case – a single roll – succeeds on any roll of target or higher. This means there are (20–target+1) success values, so it has a probability of success of (20-target+1)/20. For example, that target 8 succeeds on 8, 9, 10, … up to 20. There are (20-8+1) = 13 successful values, so it has a 13/20 = 65% chance of success.
Of course (20-target+1) can be rewritten as (21-target).
Another way of looking at it (which will become useful very soon) is that it has (target-1) failure values, and hence a (target-1)/20 chance of failure – in this case the target 8 roll fails on a 7 or lower, and so has a 7/20 chance of failure, or 35%.
Multiple roll percentages
To calculate the chance of success if you get multiple attempts is slightly counter-intuitive unless you are steeped in statistics, but I’ll try to explain.
You succeed any time a roll is successful. Therefore the only way you can fail is if all the rolls fail.
Therefore, to work out the chance of success within a certain number of rolls, you need to work out the chance of every roll failing, and then invert it.
Let’s take a target roll of 13 as an example. Based on the above, the chance of success on a single roll is (21-13)/20 = 8/20 = 40%, and the chance of failure is (13-1)/20 = 12/20 = 60%.
If you can have two attempts, the chance of both failing is 60%x60% = 36%, and hence the chance of success in one of the two rolls is (100-36) = 64%.
Note that this is the same as rolling with advantage.
If you have three attempts, the chance of failing all three is 60%x60%x60% = 21.6% and hence the chance of success in one of the three attempts is 78.4%.
You can see how quickly these chances add up!
Here’s a table of the target number on a d20 against the chance of success within a given number of attempts. This ignores 1 because you can always roll 1 or higher.
| Target number | 1 attempt | 2 attempts | 3 attempts | 4 attempts | 5 attempts | 6 attempts |
| 2 | 95.0% | 99.8% | 100.0% | 100.0% | 100.0% | 100.0% |
| 3 | 90.0% | 99.0% | 99.9% | 100.0% | 100.0% | 100.0% |
| 4 | 85.0% | 97.8% | 99.7% | 99.9% | 100.0% | 100.0% |
| 5 | 80.0% | 96.0% | 99.2% | 99.8% | 100.0% | 100.0% |
| 6 | 75.0% | 93.8% | 98.4% | 99.6% | 99.9% | 100.0% |
| 7 | 70.0% | 91.0% | 97.3% | 99.2% | 99.8% | 99.9% |
| 8 | 65.0% | 87.8% | 95.7% | 98.5% | 99.5% | 99.8% |
| 9 | 60.0% | 84.0% | 93.6% | 97.4% | 99.0% | 99.6% |
| 10 | 55.0% | 79.8% | 90.9% | 95.9% | 98.2% | 99.2% |
| 11 | 50.0% | 75.0% | 87.5% | 93.8% | 96.9% | 98.4% |
| 12 | 45.0% | 69.8% | 83.4% | 90.8% | 95.0% | 97.2% |
| 13 | 40.0% | 64.0% | 78.4% | 87.0% | 92.2% | 95.3% |
| 14 | 35.0% | 57.8% | 72.5% | 82.1% | 88.4% | 92.5% |
| 15 | 30.0% | 51.0% | 65.7% | 76.0% | 83.2% | 88.2% |
| 16 | 25.0% | 43.8% | 57.8% | 68.4% | 76.3% | 82.2% |
| 17 | 20.0% | 36.0% | 48.8% | 59.0% | 67.2% | 73.8% |
| 18 | 15.0% | 27.8% | 38.6% | 47.8% | 55.6% | 62.3% |
| 19 | 10.0% | 19.0% | 27.1% | 34.4% | 41.0% | 46.9% |
| 20 | 5.0% | 9.8% | 14.3% | 18.5% | 22.6% | 26.5% |
Obviously there is always a chance of failure, but even with a target of 6, that chance is so small within 6 attempts that it is not visible even at 3 decimal places (1 decimal place on a percentage) – the probability of success is 99.9755859375%, which rounds up to an almost certainty, as indicated by the 100% in the table.
Multiple attempts with advantage
If you have advantage, you succeed if either of two rolls succeeds. This effectively means you’re jumping along the table two attempts at a time.
| Target number | 1 attempt | 2 attempts | 3 attempts | 4 attempts |
| 2 | 99.8% | 100.0% | 100.0% | 100.0% |
| 3 | 99.0% | 100.0% | 100.0% | 100.0% |
| 4 | 97.8% | 99.9% | 100.0% | 100.0% |
| 5 | 96.0% | 99.8% | 100.0% | 100.0% |
| 6 | 93.8% | 99.6% | 100.0% | 100.0% |
| 7 | 91.0% | 99.2% | 99.9% | 100.0% |
| 8 | 87.8% | 98.5% | 99.8% | 100.0% |
| 9 | 84.0% | 97.4% | 99.6% | 99.9% |
| 10 | 79.8% | 95.9% | 99.2% | 99.8% |
| 11 | 75.0% | 93.8% | 98.4% | 99.6% |
| 12 | 69.8% | 90.8% | 97.2% | 99.2% |
| 13 | 64.0% | 87.0% | 95.3% | 98.3% |
| 14 | 57.8% | 82.1% | 92.5% | 96.8% |
| 15 | 51.0% | 76.0% | 88.2% | 94.2% |
| 16 | 43.8% | 68.4% | 82.2% | 90.0% |
| 17 | 36.0% | 59.0% | 73.8% | 83.2% |
| 18 | 27.8% | 47.8% | 62.3% | 72.8% |
| 19 | 19.0% | 34.4% | 46.9% | 57.0% |
| 20 | 9.8% | 18.5% | 26.5% | 33.7% |
Multiple attempts with disadvantage
If you have disadvantage, it’s slightly more subtle. On each attempt you only succeed if both rolls succeed. So you fail an attempt if either roll fails.
This means the chance of success is (chance of 1 roll succeeding) squared.
Above, we said the chance of one roll succeeding is (21-target)/20, so the chance of both rolls succeeding is ((21-target)/20)^2, and the chance of one of the rolls failing is 1-(((21-target)/20)^2).
This will be clearer with an example.
Returning to the target 13, remember we have a 40% chance of success and a 60% chance of failure on any particular roll.
If we have a 40% chance of success on one roll, and we need two successful rolls, we have a 40%x40% = 16% chance of success on both rolls.
If we get multiple chances, the same rules as before apply. We only fail if all attempts fail, so the chance of failure on n attempts = (chance of single failure)^n. Therefore the chance of success within n attempts = 1-(chance of single failure)^n.
Taking the example above, the chance of failure on a single attempt is 84%. So the chance of failure on two attempts is 84%x84% = 70.6%, on three attempts 84%x84%x84% = 59.3%, and so the chance of success is 16% on one attempt, 29.4% on two attempts, 40.7% on three attempts.
We said the chance of failure of a single attempt with disadvantage = 1-(((21-target)/20)^2), therefore the chance of failure in n attempts is 1-(1-(((21-target)/20)^2)^n).
Which is all getting a bit convoluted, so here it is again in a nice table.
| Target number | 1 attempt | 2 attempts | 3 attempts | 4 attempts | 5 attempts | 6 attempts |
| 2 | 90.3% | 99.0% | 99.9% | 100.0% | 100.0% | 100.0% |
| 3 | 81.0% | 96.4% | 99.3% | 99.9% | 100.0% | 100.0% |
| 4 | 72.3% | 92.3% | 97.9% | 99.4% | 99.8% | 100.0% |
| 5 | 64.0% | 87.0% | 95.3% | 98.3% | 99.4% | 99.8% |
| 6 | 56.3% | 80.9% | 91.6% | 96.3% | 98.4% | 99.3% |
| 7 | 49.0% | 74.0% | 86.7% | 93.2% | 96.5% | 98.2% |
| 8 | 42.3% | 66.6% | 80.7% | 88.9% | 93.6% | 96.3% |
| 9 | 36.0% | 59.0% | 73.8% | 83.2% | 89.3% | 93.1% |
| 10 | 30.3% | 51.3% | 66.1% | 76.3% | 83.5% | 88.5% |
| 11 | 25.0% | 43.8% | 57.8% | 68.4% | 76.3% | 82.2% |
| 12 | 20.3% | 36.4% | 49.3% | 59.5% | 67.7% | 74.3% |
| 13 | 16.0% | 29.4% | 40.7% | 50.2% | 58.2% | 64.9% |
| 14 | 12.3% | 23.0% | 32.4% | 40.7% | 48.0% | 54.3% |
| 15 | 9.0% | 17.2% | 24.6% | 31.4% | 37.6% | 43.2% |
| 16 | 6.3% | 12.1% | 17.6% | 22.8% | 27.6% | 32.1% |
| 17 | 4.0% | 7.8% | 11.5% | 15.1% | 18.5% | 21.7% |
| 18 | 2.3% | 4.4% | 6.6% | 8.7% | 10.8% | 12.8% |
| 19 | 1.0% | 2.0% | 3.0% | 3.9% | 4.9% | 5.9% |
| 20 | 0.2% | 0.5% | 0.7% | 1.0% | 1.2% | 1.5% |
You can see that, even with disadvantage, if you get six attempts, our roll with a target of 13 has more than a 90% chance of success!
Adding in the possibility of instant failure
Some schemes have three interpretations of a particular roll:
- x and above is success
- y and below is instant failure
- anything between the two is indeterminate and you can try again
For example, it is quite common to have consequences if the roll misses by 5 or more. So, with a target roll of 13, success is 13 and above, failure is 8 and below, and only a roll between 9 and 12 allows for another attempt.
How does the chance of instant failure affect the probabilities?
Taking the targets above – 13 or above for success, 8 or below for failure, 9 to 12 for another attempt:
- There is a 40% chance of success on the first roll, and a 40% chance of failure, so there is a 20% chance of being able to make a second attempt.
- On the second attempt there are the same probabilities, so the probability of success on the second attempt is 40% x 20% = 8%, giving the chance of success on the first two rolls 48%, and the chance of failure the same, so we have only a 4% chance of a third attempt.
- On that third attempt the chance of success is still 40%, so the chance of succeeding on the third attempt is 40% x 4% = 1.6%, giving a total chance of success on one of the first three rolls of 49.6%, and only a 0.8% chance of getting a fourth attempt.
Interestingly, the overall chance of success if you keep rolling (or “at the limit as n tends to infinity” as mathematicians say) is chance_of_success / (chance_of_success + chance_of_instant_failure).
In this case, the chance of success is 8 in 20, the chance of instant failure is also 8 in 20, and so if you keep rolling, once you get a result, it has a 50/50 chance of being success or failure. You can see it trending towards this in the example above.
If the target is 15 for success, that gives 10 for failure, so a 6 in 20 chance of success, 10 in 20 chance of failure, and an overall chance of success (if you could roll forever) of 6/(6+10) = 6/16 or 37.5%.
Here’s the table of chances of success if a fail by 5 or more means instant failure:
| Target | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 95.0% | 99.8% | 100.0% | 100.0% | 100.0% | 100.0% |
| 3 | 90.0% | 99.0% | 99.9% | 100.0% | 100.0% | 100.0% |
| 4 | 85.0% | 97.8% | 99.7% | 99.9% | 100.0% | 100.0% |
| 5 | 80.0% | 96.0% | 99.2% | 99.8% | 100.0% | 100.0% |
| 6 | 75.0% | 90.0% | 93.0% | 93.6% | 93.7% | 93.7% |
| 7 | 70.0% | 84.0% | 86.8% | 87.4% | 87.5% | 87.5% |
| 8 | 65.0% | 78.0% | 80.6% | 81.1% | 81.2% | 81.2% |
| 9 | 60.0% | 72.0% | 74.4% | 74.9% | 75.0% | 75.0% |
| 10 | 55.0% | 66.0% | 68.2% | 68.6% | 68.7% | 68.7% |
| 11 | 50.0% | 60.0% | 62.0% | 62.4% | 62.5% | 62.5% |
| 12 | 45.0% | 54.0% | 55.8% | 56.2% | 56.2% | 56.2% |
| 13 | 40.0% | 48.0% | 49.6% | 49.9% | 50.0% | 50.0% |
| 14 | 35.0% | 42.0% | 43.4% | 43.7% | 43.7% | 43.7% |
| 15 | 30.0% | 36.0% | 37.2% | 37.4% | 37.5% | 37.5% |
| 16 | 25.0% | 30.0% | 31.0% | 31.2% | 31.2% | 31.2% |
| 17 | 20.0% | 24.0% | 24.8% | 25.0% | 25.0% | 25.0% |
| 18 | 15.0% | 18.0% | 18.6% | 18.7% | 18.7% | 18.7% |
| 19 | 10.0% | 12.0% | 12.4% | 12.5% | 12.5% | 12.5% |
| 20 | 5.0% | 6.0% | 6.2% | 6.2% | 6.2% | 6.2% |
Note what a difference that makes, particular at the higher target numbers. With no chance of instant failure, even a target of 20 has a 26.5% chance of success after 6 rolls. With instant failure on miss-by-5, it’s only a 6.2% chance of success. Also note how the chance of success has settled to a common value (at 1 decimal place in the percentile) by the 4th roll – indicating you will almost certainly have a final result one way or the other by the 4th roll.
Critical failure
What if the chance of instant failure isn’t related to the chance of success, but instead a fixed number? The most common form of this is the critical failure – automatic failure if you roll a 1.
I won’t go through the maths. Here’s the table (ignoring advantage and disadvantage):
| Target | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 95.0% | 95.0% | 95.0% | 95.0% | 95.0% | 95.0% |
| 3 | 90.0% | 94.5% | 94.7% | 94.7% | 94.7% | 94.7% |
| 4 | 85.0% | 93.5% | 94.4% | 94.4% | 94.4% | 94.4% |
| 5 | 80.0% | 92.0% | 93.8% | 94.1% | 94.1% | 94.1% |
| 6 | 75.0% | 90.0% | 93.0% | 93.6% | 93.7% | 93.7% |
| 7 | 70.0% | 87.5% | 91.9% | 93.0% | 93.2% | 93.3% |
| 8 | 65.0% | 84.5% | 90.4% | 92.1% | 92.6% | 92.8% |
| 9 | 60.0% | 81.0% | 88.4% | 90.9% | 91.8% | 92.1% |
| 10 | 55.0% | 77.0% | 85.8% | 89.3% | 90.7% | 91.3% |
| 11 | 50.0% | 72.5% | 82.6% | 87.2% | 89.2% | 90.2% |
| 12 | 45.0% | 67.5% | 78.8% | 84.4% | 87.2% | 88.6% |
| 13 | 40.0% | 62.0% | 74.1% | 80.8% | 84.4% | 86.4% |
| 14 | 35.0% | 56.0% | 68.6% | 76.2% | 80.7% | 83.4% |
| 15 | 30.0% | 49.5% | 62.2% | 70.4% | 75.8% | 79.2% |
| 16 | 25.0% | 42.5% | 54.8% | 63.3% | 69.3% | 73.5% |
| 17 | 20.0% | 35.0% | 46.3% | 54.7% | 61.0% | 65.8% |
| 18 | 15.0% | 27.0% | 36.6% | 44.3% | 50.4% | 55.3% |
| 19 | 10.0% | 18.5% | 25.7% | 31.9% | 37.1% | 41.5% |
| 20 | 5.0% | 9.5% | 13.6% | 17.2% | 20.5% | 23.4% |
As you can see, it has a slight effect, more on the end where the chance of success is higher, and it does mean that success no longer trends towards a completely sure thing but instead towards 95%. But there is still the general trend that more attempts quickly trends towards success.
What about if anything below a 6 is an instant failure?
| Target | 1 | 2 | 3 | 4 | 5 | 6 |
| 6 | 75.0% | 75.0% | 75.0% | 75.0% | 75.0% | 75.0% |
| 7 | 70.0% | 73.5% | 73.7% | 73.7% | 73.7% | 73.7% |
| 8 | 65.0% | 71.5% | 72.2% | 72.2% | 72.2% | 72.2% |
| 9 | 60.0% | 69.0% | 70.4% | 70.6% | 70.6% | 70.6% |
| 10 | 55.0% | 66.0% | 68.2% | 68.6% | 68.7% | 68.7% |
| 11 | 50.0% | 62.5% | 65.6% | 66.4% | 66.6% | 66.7% |
| 12 | 45.0% | 58.5% | 62.6% | 63.8% | 64.1% | 64.2% |
| 13 | 40.0% | 54.0% | 58.9% | 60.6% | 61.2% | 61.4% |
| 14 | 35.0% | 49.0% | 54.6% | 56.8% | 57.7% | 58.1% |
| 15 | 30.0% | 43.5% | 49.6% | 52.3% | 53.5% | 54.1% |
| 16 | 25.0% | 37.5% | 43.8% | 46.9% | 48.4% | 49.2% |
| 17 | 20.0% | 31.0% | 37.1% | 40.4% | 42.2% | 43.2% |
| 18 | 15.0% | 24.0% | 29.4% | 32.6% | 34.6% | 35.8% |
| 19 | 10.0% | 16.5% | 20.7% | 23.5% | 25.3% | 26.4% |
| 20 | 5.0% | 8.5% | 11.0% | 12.7% | 13.9% | 14.7% |
I hope it will be obvious in this case that a target below 6 has the same chances as a target of 6.
Now, with a 25% chance of instant failure, the chances ramp up more slowly, and ultimately trend towards a 75% chance of success rather than 100%.
Take 10 and Take 20
Third edition D&D had the concepts of “take 10” and “take 20”.
Take 10 – “When your character is not being threatened or distracted, you may choose to take 10. Instead of rolling 1d20 for the skill check, calculate your result as if you had rolled a 10. For many routine tasks, taking 10 makes them automatically successful. Distractions or threats (such as combat) make it impossible for a character to take 10.”
Taking 10 is implicitly a single attempt, and so doesn’t really play into the multiple attempt tables. You would only take 10 when you think the target number is 10 or lower.
Take 20 – “When you have plenty of time (generally 2 minutes for a skill that can normally be checked in 1 round, one full-round action, or one standard action), you are faced with no threats or distractions, and the skill being attempted carries no penalties for failure, you can take 20. In other words, eventually you will get a 20 on 1d20 if you roll enough times.”
This is very different. This is making the speed at which the chance improves with multiple attempts explicit, and stating that, even with the most difficult task you could attempt (i.e. needing a 20 on d20), if you continue trying, eventually you will roll a 20.
It talks about 2 minutes. Given that a single round is taken to be 6 seconds, that is 20 rounds. How does that fit with the actual probabilities? Extending the single-roll table to 10 rolls (1 minute), 20 rolls (2 minutes), 30 rolls (3 minutes), 40 rolls (4 minutes), 50 rolls (5 minutes) and 60 rolls (6 minutes) we get:
| Target | 10 attempts | 20 attempts | 30 attempts | 40 attempts | 50 attempts | 60 attempts |
| 2 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 3 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 4 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 5 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 6 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 7 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 8 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 9 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 10 | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 11 | 99.9% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 12 | 99.7% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 13 | 99.4% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 14 | 98.7% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |
| 15 | 97.2% | 99.9% | 100.0% | 100.0% | 100.0% | 100.0% |
| 16 | 94.4% | 99.7% | 100.0% | 100.0% | 100.0% | 100.0% |
| 17 | 89.3% | 98.8% | 99.9% | 100.0% | 100.0% | 100.0% |
| 18 | 80.3% | 96.1% | 99.2% | 99.8% | 100.0% | 100.0% |
| 19 | 65.1% | 87.8% | 95.8% | 98.5% | 99.5% | 99.8% |
| 20 | 40.1% | 64.2% | 78.5% | 87.1% | 92.3% | 95.4% |
From this you can see that even after 60 attempts (6 minutes) of trying to roll a 20 – three times the take 20 “plenty of time”, you still have a 4.6% chance of failing! And at the 2-minute mark, even with a target of 18, it’s not a completely sure thing – a 3.9% chance of failure, and with a target of 20, you’ve only about a 2 in 3 chance of succeeding. So it turns out this is quite generous.
Summary
There has been a lot of detail here. What can we take from it?
First, if there is no cost to multiple attempts, the chance of success increases rapidly with each attempt, particularly in the middle and lower portions of the table (target 15 and below).
This is why I generally don’t allow multiple attempts at my table, and don’t generally allow another player to make the same check when they find that someone fails. Or I try to come up with some cost to continuing to check. At a minimum, the time/noise of the check increases the chance of a complication wandering in. If you’re not already using tension dice, I strongly recommend it – add a die to the pool for every check.
The temptation of everyone piling in is where the Help action comes in – rather than multiple checks, others in the party can participate in the check by giving the primary roller advantage. I always insist that the helper has some reason why they would be able to help, though – for example, also having proficiency in a skill, and actually being able to get close enough to help. And they have to declare they’re helping before the check actually happens.
That’s if there’s no cost to retrying. If you introduce a chance of instant failure, the odds of success ramp more slowly, and you quickly trend towards a final result one way or the other. The overall chance of success over multiple rolls depends on the chance of instant success compared with the chance of instant failure, ignoring the chance of the reroll.
I will use this analysis when comparing alternative lock-picking strategies to help quantify “this feels too easy”, “this feels too hard” and “this feels about right.”
Thanks for the cool tables. I actually enjoy the probability maths behind RPGs so this was a fun read for me. Am I weird? Probably. Do I care? Not too much. 😀