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Say I collect 40 perfectly random integers between 1 and 400. What's the chance that any integer is repeated consecutively six times in such a random draw?

What I'm looking for is the chance of sequences like [372, 193, 42, 42, 42, 42, 42, 42, 274, 42, 7, ...], [372, 193, 42, 42, 42, 42, 42, 42, 274, 242, 7, ...], or [372, 193, 42, 42, 42, 42, 42, 42, 42, 42, 42, ...] as they all fulfills what I'm looking for. As a counter example [372, 193, 42, 42, 42, 42, 42, 77, 274, 42, 7, ...] does not satisfy my conditions because the six 42's are not consecutively repeated.

The Birthday problem gives that it's an 87% chance that two of the 40 are the same number but I'm failing to go from that to calculating the chance of a certain integer consecutively repeating itself n number of times in the random collection.

  • 0
    When you say "number", do you mean integer or real number? When you say "random", according to what distribution?2011-06-29
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    Integer and lets imagine a perfectly random distribution.2011-06-29
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    Before computing the probability of an event you must carefully define it: `the same number repeats itself six times` is bad formulated. You mean that some (any) number appears exactly six times? Regardless of that some other number appears more that six times (or six times)?2011-06-29
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    Any of the integers between 1 to 400 appearing after another six times. Like `[301, 23, 42, 42, 42, 42, 42, 42, 255, 120, 42, ...]`. This is six times repeated. I will try to clarify this in the question.2011-06-29
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    Must it be repeating itself exactly six times (of the 40 random numbers)? (i.e., appears exactly 7 times)? or at least 7 times?2011-06-29
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    @amWhy It has to, somewhere in the random set, be `[x, y, n, n, n, n, n, n, z, i]`. Six `n`s in a row and I guess that that could be expressed as five repeats.2011-06-29
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    So, you are looking for the probability of selecting 40 integers from a pool of 400 such that a some number is selected 6 consecutive times? (And it is okay if it appears again later)...Exactly 6 consecutive repetitions, or at least 6 consecutive repetitions? i.e. x y n n n n n n x q ... works, but does x y z n n n n n n n n a b c ... also count?2011-06-30
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    You might want to edit your question, addressing questions asked in these comments. (e.g. using consecutive vs repeating (or repeating consecutively instead of only repeating, exactly 6 times vs. at least 6 times...etc.2011-06-30
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    @amWhy `from a pool of 400` - no, not a pool. Is it more clear if I call it a collection of 40 random integers between 1 and 400?2011-06-30

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