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There's this question: You roll a die three times, what's the probability of getting 2 sixes in a row?

I have a solution for the problem that says: either getting 2 sixes in a row (first pair) or getting 2 sixes in a row (last pair), or getting 3 sixes in a row.

Since the first two cases get: 1/6 * 1/6 * 5/6 * 2 Then the last case: 1/6 * 1/6 * 1/6 This gives a probability of 11/216.

My first try on this exercise I wanted to use combinatorics.

My intuition led me to calculate all the possible outcomes which is 6^3 = 216 (6 numbers possible, 3 times)

This gives the denominator. Yet I've been stuck to find the number of favorable outcomes (nominator) using combinatorics. I understand that it is:
(6,6, 1–5) which gives 5 combinations or
(1—5, 6,6) which gives again 5 combinations or
(6,6,6) which is 1 combination

this gives a probability of 11/216.

But I can't find a mathematical formula that fits, and in a case where the combinations are too big, I wouldn't have found it.

Can someone tell me the way to solve this using combinatorics formula for any n and p?

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  • $\begingroup$ You already used combinatorics! You seem to have a generalization of this problem in mind, but there are many different ways it could be generalized. What is yours? $\endgroup$
    – whuber ♦
    Commented Jan 19, 2022 at 19:19
  • $\begingroup$ I wanted to generalize this problem to any n number of rolls. What would be the probability to get p consecutive outcomes. For example, throw the dice 10 times, what is the proba of getting number "5" 6 consecutive times. $\endgroup$
    – Bouji
    Commented Jan 19, 2022 at 19:29
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    $\begingroup$ Okay: so please state that in your post itself. I suspect we might already have a thread that answers this: your question is about the chances of runs in a series of coin flips. (The "coin" in your case has a $1/6$ chance of landing on the chosen number and $5/6$ chance of not.) stats.stackexchange.com/questions/332600/probability-of-runs asks the same question, but has no effective answer (yet). stats.stackexchange.com/questions/362470 has two computing answers but no formulas. stats.stackexchange.com/questions/158490 is related, with a good answer. $\endgroup$
    – whuber ♦
    Commented Jan 19, 2022 at 21:44

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