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Probability of Consecutive Dice Rolls
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What is the probability of getting at least 3 consecutive sixes when a die is thrown 5 times?

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It can be divided into 3 cases

Case 1 : Possibilities for only three consecutive 6's but no 4 or 5 consecutive 6's:

The sequence of 3 consecutive 6's can be one of below three types:
666XY,X666X,YX666
where the X's can be choosen from (1,2,3,4,5) and the Y's are allowed from (1,2,3,4,5,6)

So the total number of possibilities for only 3 consecutive 6's are
666XY = 5x6 = 30
X666X = 5x5 = 25
YX666 = 6x5 = 30

Total 85 such possibilities

Case 2 : Possibilities for four consective 6's but no 5 consecutive 6's:
6666X and X6666
where the X can be choosen from (1,2,3,4,5)

So there are 5+5 = 10 such possibilites.

Case 3 : Possibilities for 5 consecutive 6's
There is only one possibility for 5 consecutive 6's : 66666

So the total probability of atleast 3 consecutive 6's are:

p = Favourable outcomes / total number of outcomes
p = 30+25+30+10+1 / 6x6x6x6x6
p = 96 / 6x6x6x6x6
p = 16 / 6x6x6x6
p = 1 / 81

So the probability of getting at least 3 consecutive sixes when a die is thrown 5 times is 1/81

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