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There's this question: You roll a dice three times, what's the probability of getting 2 "6" in a row?

I have a solution for the problem that says: either getting 2 six in a row (first pair) or getting 2 six in a row (last pair), or getting 3 six 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 exercice 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, 1to5) which gives 5 combinations or (1to5, 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

Bouji
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  • 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? – whuber Jan 19 '22 at 19:19
  • 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. – Bouji Jan 19 '22 at 19:29
  • Okay: so please state that in your post itself. I suspect we might already have a thread that answer 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.) https://stats.stackexchange.com/questions/332600/probability-of-runs asks the same question, but has no effective answer (yet). https://stats.stackexchange.com/questions/362470 has two computing answers but no formulas. https://stats.stackexchange.com/questions/158490 is related, with a good answer. – whuber Jan 19 '22 at 21:44

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