working out % chance?

[machine]

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If the chances of being picked in each show is 1/22 (if I'm thinking of the right show it is as they replensih the numbers each time),
Chances of being picked in first show are 1/22
Chances of being picked in second show are 1/22
etc.
Chances of being being picked by first show are (1-21/22) = 2/22 = 4.5%
Chances of being being picked by second show are 1-(21/22 *21/22) = 8.9%
Chances of being being picked by third show are 1-(21/22 *21/22*21/22) = 13.1%
etc.
Or to formalise it 1-(21/22)^X where x is the number of shows.
If the number of people left that are up to qualify goes down then it's slightly different.

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this looks right, picks are random and numbers are replenished.

anyone care to workout the probability of not being picked after 20 shows?