What is probability of pattern “number appears-none-appears” to happen in a lottery?

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codeuniv

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What is probability of pattern “number appears-none-appears” to happen in a lottery?

In a lottery 5 out of 36 (5 numbers drawn from [1..36] without replacement) what is the probability of same number (say, 13 - but logic applies to any number) to appear in this pattern:
start of game:

  • (possibly many plays, where pattern "play1-play2-play3" not observed, but maybe this pattern happens from the first game - we do not care for this)
  • play 1: appears (example: 13 7 11 8 30 )
  • play 2: none (example: 36 17 19 1 33 )
  • play 3: appears (example: 32 15 13 6 2)
  • ... (many more plays)
  • play N, N+1, N+2: same pattern (as in plays 1,2,3) again (according to probability of this pattern appearance)
Purpose why I am asking - I am planning to monitor a real lottery and see if real frequency of this pattern occurencies differs much from theoretical (defined in this question by formula above).

I have an idea how to solve it - here on mathoverflow, but I am not sure in my solution.
 
codeuniv said:
In a lottery 5 out of 36 (5 numbers drawn from [1..36] without replacement) what is the probability of same number (say, 13 - but logic applies to any number) to appear in this pattern:
start of game:

  • (possibly many plays, where pattern "play1-play2-play3" not observed, but maybe this pattern happens from the first game - we do not care for this)
  • play 1: appears (example: 13 7 11 8 30 )
  • play 2: none (example: 36 17 19 1 33 )
  • play 3: appears (example: 32 15 13 6 2)
  • ... (many more plays)
  • play N, N+1, N+2: same pattern (as in plays 1,2,3) again (according to probability of this pattern appearance)
Purpose why I am asking - I am planning to monitor a real lottery and see if real frequency of this pattern occurencies differs much from theoretical (defined in this question by formula above).

I have an idea how to solve it - here on mathoverflow, but I am not sure in my solution.
Click to expand...

If there is anything you can observe that makes it more predictable, the sponsors have failed miserably.
 
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