This strikes me as an interesting (though potentially tricky) lottery question that I'm having trouble getting my head around.
Many state lotteries provide 'ball history' tables on their websites listing each number and the draws that have elapsed since it last came out. Players then tend to categorise them based on how 'hot', 'cold', 'fast' or 'slow' they were, or apply trend analysis techniques in an attempt to predict future outcomes and strike it rich.
However, I'm only interested in the 'mechanics' driving these things. Supposing we had a game where 6 balls are drawn from 45 available balls, the game is drawn weekly, and the lottery has been running for 52 weeks.
We ignore the very first drawing (nothing to measure up to this datum if you like) and start analysing from the second week. So, given that all balls are returned to the barrel after each draw:
(1) how many balls in total can we expect to see re-appear the very next week over the space of one year?
(2) how many can we expect to see drawn, skip one week and appear the following week?
(3) get drawn, skip two weeks then re-appear, and so on?
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