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If I have an independent event E with two possible outcomes x and y with their own probabilities p(x) and p(y) and I then repeat event E a certain number of times (n), how can I calculate the probability of that sample of n events containing at least one sequence of consecutive identical results at least L occurrences long?

So for example, if I roll a fair 6 sided die each morning and consider it a "win" when I roll 5 or 6, how do I calculate the chances that if I do this for a year, I will have at least 9 "wins" in a row at some point?

So in my above example:

E: Rolling a fair six-sided die
x: Rolling 5 or 6
y: Rolling 1, 2, 3 or 4
p(x): 1/3
p(y): 2/3
n: 365
L: 9

Can I show for any values of these variables how to determine the probability of there being at least one instance of L consecutive occurrences of x within my n quantity of trials of E?

3N1GM4
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  • Easiest is probably to do it recursively. Your sequence must begin with one of $y, xy, x^2y, \cdots, x^L$ where $y$ is anything other than a "win". If you get $x^L$ at the start you can stop, with any of the others you restart with a shorter sequence. – lulu Jan 10 '20 at 16:26
  • Thank you yes, that other question does look like it is the same, apologies for the dupe. I will also take a look at the recursive solution suggested by lulu. – 3N1GM4 Jan 10 '20 at 16:35

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