Probability of i event_A and j event_B on n dice

I am trying to figure out probability for a game using 6-sided dice. The basic idea is that the player rolls n dice and gets a success for each 6 rolled. So far, so simple; I'm using Python's scipy.binom.pmf function which gives the probability of rolling i successes on n dice with a given chance of success, 1/6 in this case.

But the game offers a "bonus". With the bonus, in addition to 6s being successful, one 5 in the roll is also a success. (So a roll of 2, 3, 5, 5, 6, 6 would be two successes normally, but three with the bonus.)

How can I compute the probability of i successes on n dice with this bonus condition?

• It is unclear to me when the "bonus" applies. Is it decided before the roll (like "roll with advantage" in D&D), or is it something that applies based on the outcome of the roll (like "if there's a natural 6, then one 5 can count as a success as bonus")? Feb 2 '21 at 1:26
• Do yu mean that m**exactly-- one 5 is a bonus success, ort at least one 5? How many successes is a roll of 6 6 5 5 4 3? Feb 2 '21 at 1:37
• I know your using python but there is a useful website here anydice.com. You can work out many dice probability there and with a bit of coding create some complicated combinations and rules. Feb 2 '21 at 10:07
• Are you looking for exactly i successes or at least i successes? Feb 2 '21 at 12:58
• @DavidSiegel OP already provides an example which is equivalent to yours and answers your question. Feb 2 '21 at 15:27

P(i successes) = ((# of ways to get i 6s and no 5s) + (# of ways to get i-1 6s and some 5s)) / 6^n

(# of ways to get i 6s and no 5s) = (n choose i) * 1^i * 4^(n-i) [pick which dice must be 6s, 1 way for each of those to be a 6, 4 ways for each of the others to be neither 6 nor 5]

(# of ways to get i-1 6s and some 5s) = (n choose i-i) * 1^(i-1) * (5^(n-(i-1)) - 4^(n-(i-1))) [pick which dice must be 6s, 1 way for each of those to be a 6, 5 ways for each of the others to be non-6 except then we're overcounting by the 4^(n-(i-1)) ways that they are ALL neither 6 nor 5]

Dropping the unnecessary powers of 1, overall your probability can be computed exactly as: [Note: I am using the convention that (n choose -1)=0, unlike Python's math.comb which does not allow negative inputs.]

• If you haven't seen "choose" before, see en.wikipedia.org/wiki/Combination. It's math.comb in Python Feb 1 '21 at 23:41
• @John Thanks for the pretty picture :) Feb 2 '21 at 21:24
• @Benjjamin Cosman, I think I see what you're getting at, and I think you are on the right track, but there's still something wrong or I'm not interpreting you right. Here's the Python code I've got: Feb 3 '21 at 0:15
• pSixes = comb(n, i) * pow(1/6, i) * ( pow(4/6, n - i)) pBonus = comb(n, i-1) * pow(1/6, i-1) * (1 - pow(4/6, n - (i - 1))) result[i] = pSixes + pBonus For 3 dice, that produces P(1 success) = 0.925925925925926, P(2 successes) = 0.33333333333333337, and P(3 successes) = 0.032407407407407406, ... Feb 3 '21 at 0:20
• Oops; fixed. I've now checked that this one gives the same answer as @StartPlayer (and adds up to 1 :) ) Feb 3 '21 at 16:20

I used [Any Dice] website to work this out.

The code below is probably not the cleanest but I only learned this site today. The function below will count 1 success if at least one 5 and add to that how many sixes rolled. This is getting results of to 5d6. function: contains VALUES:s in SEQUENCE:s {
FIVES: 0
SIXES: 0
loop P over {1..#VALUES} {
if (5 = SEQUENCE) {
if FIVES = 0 {
FIVES: 1
}
}
SIXES: SIXES + (P@VALUES = SEQUENCE)
}
result: FIVES + SIXES
}

output [contains { 6 } in d6]
output [contains { 6 } in 2d6]
output [contains { 6 } in 3d6]
output [contains { 6 } in 4d6]
output [contains { 6 } in 5d6]

These are the results that I've formatted to be a bit more readable than the anydice website gives me. If you roll 1d6 the chance of 1 success (A 6 or 5) is 33.3%

If you roll 3d6. the chance of 2 successes is 18.05% (that's either 2 6s and no 5s or one 6 and at least one 5)

1d6
0 success, 66.6666666667
1 success,33.3333333333

2d6
0 success, 44.4444444444
1 success, 47.2222222222
2 success,8.33333333333

3d6
0 success, 29.6296296296
1 success, 50.462962963
2 success, 18.0555555556
3 success, 1.85185185185

4d6
0 success, 19.7530864198
1 success, 48.225308642
2 success, 26.2345679012
3 success, 5.4012345679
4 success, 0.385802469136

5d6
0 success, 13.1687242798
1 success, 43.4799382716
2 success, 31.9573045267
3 success, 9.90226337449
4 success, 1.4146090535
5 success, 0.0771604938272

• Thanks for that! I wrote the 3 dice case manually, and it matches with your results. The problem is that once you get up to 8 or 9 dice, it's too slow (and I've forgotten how to get multiple dice loops :-)). Feb 3 '21 at 0:24
• The 1 dice case was clearly correct and I drew about a 6x6 grid to check 2d6 results were correct as well. An advantage of this site is you can write code to check and kind of odds. if you have a bag or 7 tokens and two are "success" you can work that out as well by saying d7 as a 'dice'. It takes a bit of coding and the site docs seem clear enough. Feb 3 '21 at 9:47