Question concerning probablity

[Mins]

How can I calculate the probablity of 2 same numbers being right next to each other, when 100 random numbers chosen from '1,2,3.... 6' form a line?

[CRGreathouse]

That's pretty well certain. To five decimal places, S = 100%.


To see why, and to get a more precise answer:
Let S denote a success, that is at least two in a row are the same. Consider instead the smaller problem with only two such dice in a row:

(1-6)(1-6)

S is just the chance that the two are the same, which is 6/36 = 1/6.

For three dice:

(1-6)(1-6)(1-6)

The middle die has a 1/6 chance of being the same as the one before it, and the last die has a 1/6 chance of being like the die before it. That's 1/6 + 1/6, except that now you're double counting when all three are the same, so it's 1/6 + 1/6 - 1/36.

This is easier if you calculate 1-S, which is 5/6 with two dice and (5/6)^2 for three dice.

[Mins]

This is easier if you calculate 1-S, which is 5/6 with two dice and (5/6)^2 for three dice.

How can you calculate the probability of 'n' pairs of same numbers being right next to each other?

[CRGreathouse]

How can you calculate the probability of 'n' pairs of same numbers being right next to each other?

I'm not sure what you mean, give an example.

[Mins]

I'm not sure what you mean, give an example.

10 numbers will be chosen from 1,2,3...6, and they will form a line, like this

1442345662

There are 2 pairs of sets, as underlined above.

What I want to find out is the probability of not just only one, but 2 or more sets(2 same numbers being right next to each other) appearing in a line.

[CRGreathouse]

What I want to find out is the probability of not just only one, but 2 or more sets(2 same numbers being right next to each other) appearing in a line.

Does 1112345 count as having two "sets"? Does 1111234?

[Mins]

Does 1112345 count as having two "sets"? Does 1111234?
First one has one "set"(I defined the word, think you should know).
And second one has two "sets".
But if ruling those out simiplify your calculation, think it's OK.

[Mins]

What I want is a method to calculate those :

two sets appearing in a line of 50 numbers
three sets appearing in a line of 50 numbers
four sets appearing in a line of 50 numbers
etc.

But a solution that can be used in similar situations will help me.