By asking my players to match my roll, am I making their checks harder?

Question

Sometimes I ask my players to make a roll to see if something random happens (winning a game of chance, for instance, or a wilderness encounter). These rolls won't involve adding any modifiers, they'll almost always be straight up or down rolls.

In order to preserve some mystery, I won't tell them beforehand what they have to roll in order to achieve the outcome and I'll assign a random number or range of numbers to the outcome. Primarily this is to try and move away from the 20 = good and 1 = bad mentality that my players tend to have when rolling. It also means that if I make them roll a d20 when their characters are settling down to camp overnight, even though my players might have read the DMG, it won't actually be a roll of 17-20 that results in a random encounter, and instead it'll be a roll of 4-7 instead (for instance). Keeps them on their toes.

So what I've been doing is rolling a d20 in secret and then seeing if the player rolls a matching number. If I'm looking for a range I'll just add the size of the range to my rolled number and wrap around. E.g. I roll an 18. For an event with 25% chance of occurrence I'll look for the player to roll an 18, 19, 20, 1 or 2.

From a statistics perspective the odds of rolling any particular number on a d20 is 1/20. But to roll two of the same number in a row would be 1/400, which is much less likely.

So my question is: by expecting players to match the roll of my dice instead of picking a number at random (or sticking to the suggested ones in the DMG) am I inadvertently making it less likely for them to succeed?

Answer 1

No.

From a statistics perspective the odds of rolling any particular number on a d20 is 1/20. But to roll two of the same number in a row would be 1/400, which is much less likely.

You are right that the chance of rolling \$\{20, 20\}\$ is 1/400, for example. But you actually have 20 possible outcomes that will result in a success for the player, i.e., if you roll 1 and he rolls 1, he succeeds, if you roll 2 and he rolls 2, etc... So, the actual probability of success \$P(s)\$ is given by

$$ P(s) = \sum_{d = 1}^{X} P(d)^2 $$

where \$X\$ is the dice size (i.e. 20 for a d20) \$P(d)\$ is the probability of a specific dice value being rolled (1/X), in the case of the d20, simply 1/20. Since they are uniformly distributed and, by definition of a probability, the sum has to be 1, it goes that

$$ P(s) = \frac{1}{X} \sum_{d=1}^{X} P(d) = \frac{1}{X}$$

exactly equal to the probability of any dice number being rolled.

Another way to see it is a conditional probability: the probability of he rolling a number Y given that you already rolled Y is 1/X.

Or, skipping the math and answering it: no, you don't change the probability at all. It stays exactly and perfectly the same.

Answer 2

No, but maybe yes.

HellSaint is right about the math. As long as the number of possible outcomes that are a "success" is the same, it doesn't matter which numbers they are.

However, in any situation where they would get to add their characters' stats or bonuses to the roll, this method would negate the benefit of doing things the character is good at. If you want to leave it up to pure chance that's fine, but then you might as well just roll the die yourself.

Take the chance of a wilderness encounter. Is that really pure luck, or does a seasoned explorer have some advantage? Is there anything they can do--post more guards, don't light a fire, get a watchdog--to be less likely to get a hostile encounter or more likely to get a helpful one? With a straight high roll, you can give bonuses, which rewards your players for having good ideas and for playing to their party's strengths. With the method you're using, you could expand the range of success numbers, but the players don't see it happening, so they won't feel rewarded.

In general you shouldn't make your players roll for success or failure on a task that they have no control over. Even if their characters are in a gambling den and the die roll represents a literal die roll, why have them gamble and not give them the option to cheat?

Answer 3

You said:

From a statistics perspective the odds of rolling any particular number on a d20 is 1/20. But to roll two of the same number in a row would be 1/400, which is much less likely.

This is an incorrect analogy.

While the chances of rolling a particular (e.g. "3") twice in a row is indeed 1/400 for a d20, that is not what you're asking them to do. You're asking them to roll the same number you rolled.

One way of looking at this is to realize that of the 400 possible combinations that come up, 20 of them are valid hits - if you roll 1 and they roll 1; if you roll 2 and they roll 2; and so forth. If 20 of the 400 are hits, then the chances of a hit is 1/20.

The other way of looking at this is that your die roll is GUARANTEED to come up with a number; any number is valid. After you have rolled, there is a 1/20 chance their roll comes up with the same number.

Of course, if both of you roll the same die, and it isn't balanced, or if your two dice are unbalanced in the same way, this will increase their chances (of course, if they are unbalanced in different ways, it will work against them).