My friends are in a bit of an argument over Dungeons & Dragons.
My player managed to guess the outcome of a D20 roll before it happened, and my friend said that his chance of guessing the number was 1 in 20. Another friend argues that his chance of guessing the roll is 1 in 400 because the probab...
If you roll 2d20, what's the probability that both dice show the same number? (Alternatively, consider the case of 2d3, which is conceptually the same, but easier to enumerate.)
@Nelson My comment isn't intended as an answer, but a way for OP to start thinking through how to answer their own question. This is the preferred way to address questions which might appear on homework problem sets or exams; more information can be found in the self-study wiki. Moreover, it's hard to understand how a question can be "wrong."
The introduction of 2 dice isn't accurate representation of what is happening. The probability of the first "pick" is technically 100%, because it is impossible for the first action to not produce a result, hence 100%. It is only the second action that has a probability of failure (not matching the first pick). There is no chance of failure for the first action of rolling a dice. You just need to end up with a number between 1 and 20.
So if you don't guess the roll, and always pick 1 (or whatever is your lucky number), then your chances of having a correct guess suddenly increase from 1/400 to 1/20.
@Nelson Trying to get a match on 2d20 seems like a pretty accurate representation of what's happening. With the guess, the number chosen by an intelligent guesser has a 100% chance of being a valid result, and the roll has a 1/20 chance of matching. With the 2d20, the first number rolled has a 100% chance of being a valid result, and the second roll has a 1/20 chance of matching.
@SextusEmpiricus LOL, no, this is why the moment 2d20 is mentioned, it throws people off. The probability of getting a specific number using both dice is 1/20 * 1/20 = 1/400, but having the two dice match is only 1/20 because the first die's probably doesn't affect the second die, and if it is a matter of matching, two die matching each other is the same probability as one die matching a specific number (1/20). When you try to have both die match a specific number, the first die needs to match (1/20) AND the second die needs to match (1/20 * 1/20).
Many of the answers given rely on the assumption that the guesser is equally likely to guess each of the 20 numbers. But there is nothing in the question to indicate that this is true, and in real life most humans are nowhere near being perfect random number generators.
Assuming the roll was behind a DM screen, had already been made but not revealed, but the DM's expression was animated I'd say it's likely the chances of guessing are 50/50. Which is to say, if the DM rolled a 1 or a 20 their reaction would be more pronounced, making it easier to guess what they roll then if they'd rolled a middling number. Not an answer since this involves being able to "read" another person, and also assumes the die has been rolled but the exact result hasn't been revealed.
Imagine the DM had a hidden number that was a required roll in order to progress. The chance that the player guesses that number (randomly) and then the die also rolls that number is 1/400. However, just the probability that the die matches the player’s guess is 1/20. Hope this helps.
@Sycorax so did I convince you that the answers which rely on the assumption that all numbers are equally likely to be guessed don't properly answer the question?
@fblundun I don't think it's incorrect to answer the question with an explicit assumption that a person can choose a number uniformly at random. If we have it your way, then we also have to consider whether the specific dice used in this D&D game are fair.
@Sycorax I don't buy that at all - dice are manufactured to be very close to fair, but human guesses are nowhere near uniformly distributed, and there's no suggestion in the question that the guesser was even attempting to guess randomly.
Except the dice that are poorly manufactured, made from nonuniform materials (lots of D&D dice are made from exotic or organic stuff these days), or expressly manufactured to be biased. Perhaps the reason a player decided to guess the die result in the first place was because these dice were consistently showing a specific value.
It's strange to decide on the one hand that all questions about a person's ability to select numbers randomly are in scope while at the same time defending that all questions about the fairness of the dice are out of scope. If you further make additional assumptions, then of course we can pick and choose -- but that's what I'm saying in the first place: make your assumptions explicit.
@Sycorax I don't agree - in probability questions it's conventional to assume that dice are fair unless the question explicitly raises the possibility that they aren't (since dice exist specifically to generate random numbers fairly). It isn't conventional to assume that a human picking a number would do so uniformly at random from their choices.
The answer to "What is the probability of rolling an odd number on a d20" is "0.5"; the answer to "If I ask a man on the street to pick a number between 1 and 20, what is the probability he will pick an odd number?" is much more complicated.
The question includes reasoning about 1/20 probability of a person guessing a number. Why are you insisting that your beliefs about humans’ randomness overrides what the question says?
Discussion on question by Theguy What…
Discussion on question by Theguy What…