how to calculate the odds of 5d6 higher than 1d10?

Question

In anydice or troll, how can we calculate the odds of rolling 5d6 and counting the successes against 1d10? Successes are every result in each six-sided die equal or higher than the d10 result.

This means that, if the d10 rolls 7, 8, 9, or 10, you always have zero successes, and if it rolls a 1, you always have 5.

I've tried [count {1..d10} in 5d6], but anydice says sequence must end with a number. I've tried 5d6>1d10, but obviously anydice takes 5d6 as the sum of the dice. Maybe it can be done with a custom function, but that seems beyond my skills...

Answer 1

You need to cast your dice to a sequence

The trick for dealing with a roll of dice as individual rolls (and not their sum) is to cast it into a sequence. For this, we use a function and specify our dice parameter as a sequence (appending :s). Once we have it as a sequence, we can compare each value to a number. That will give the number of elements in the sequence for which the comparison is true.

function: number of A:s greater or eq than T:n {
   result: A >= T
}

output [number of 5d6 greater or eq than 1d10]

Answer 2

Here's probably the fastest way to do it:

function: test T:n {
  X: d6 >= T
  result: 5dX
}
output [test d10]

The first trick used here is that, when a die (here, a d10) is passed to a function expecting a number (i.e. as a parameter tagged with :n), AnyDice will automatically call the function for every possible sum of the roll and calculate a weighted average of the results. Thus, inside the function the roll is effectively "frozen" and can be treated as a fixed number.

Next, I use the expression d6 >= T to obtain a distribution that equals 1 with probability \$(7-T) \mathbin/ 6\$ (i.e. the probability of rolling at least T on a d6) and 0 otherwise, and save it as a custom die in the variable X. And then I roll 5 of these custom dice to get the number of rolls exceeding T on 5d6 and return it.

Of course, if you want to make the function more flexible, you could also pass the die to be rolled and/or the number of dice in the pool as additional parameters to it.

Answer 3

This should work...

function: csuccess roll ROLL:s with target TARGET:n {
  if TARGET < 7 {
    result: [count {TARGET..6} in ROLL]
  }
  else {
    result: 0
  }
}

output [csuccess roll 5d6 with target 1d10]

For a better understanding of what this is doing, this is a less efficient method of the exact same solution:

function: csuccess roll ROLL:s with target TARGET:n {
  NUMONES: [count {1} in ROLL]
  NUMTWOS: [count {2} in ROLL]
  NUMTHREES: [count {3} in ROLL]
  NUMFOURS: [count {4} in ROLL]
  NUMFIVES: [count {5} in ROLL]
  NUMSIXES: [count {6} in ROLL]

  if TARGET = 1 {
    result: NUMONES+NUMTWOS+NUMTHREES+NUMFOURS+NUMFIVES+NUMSIXES
  }
  if TARGET = 2 {
    result: NUMTWOS+NUMTHREES+NUMFOURS+NUMFIVES+NUMSIXES
  }
  if TARGET = 3 {
    result: NUMTHREES+NUMFOURS+NUMFIVES+NUMSIXES
  }
  if TARGET = 4 {
    result: NUMFOURS+NUMFIVES+NUMSIXES
  }
  if TARGET = 5 {
    result: NUMFIVES+NUMSIXES
  }
  if TARGET = 6 {
    result: NUMSIXES
  }
  result: 0
}

output [csuccess roll 5d6 with target 1d10]