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

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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