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Consider the following scenario:

Suppose on some date $D1$ the number $N$ is a winning number in a fair lottery where a "play" is considered the selection of a finite set of numbers. By "fair" I mean that the winning number will be selected at random. At some later date $D2$ another lottery, using the same rules, will be held. Suppose one picks the same number $N$ that was the winning number on $D1$ to play on $D2$. Does this selection increase/decrease or have no effect on ones chances of winning the lottery on $D2$?

I believe that picking a previously winning number will have no impact on one's chance of success for the simple reason that the universe has no way of remembering which previous numbers won and which ones did not; since the selection is random, each number has an equally likely chance of being picked regardless of whether it was picked before. Other than a basic assumption of causality, I really don't know though how one would rigorously prove this.

The counterargument, which I believe is faulty, argues against "reusing" winning numbers because the likelihood of the same number coming up twice is infinitesimally small so, of course, one should not reuse numbers. The problem with this as I see it though is that the probability of picking any two specific numbers, regardless of whether they are the same, is identical to picking the same number twice. The fault is that picking numbers this way is like trying to play two lotteries in succession - which is very different from the given problem.

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Your analysis is correct, in situations where one wins a fixed amount of money if one matches a winning number. There will be no effect. Or at least there will be no effect under the usual assumptions of independence, for which there is strong empirical evidence. If the lottery picks from say the first $100$ natural numbers, the sequence $47$ this week, $17$ next week is just as likely as the sequence $17$ this week, $17$ next week. Each is quite unlikely (probability $1/10000$). The second is likely to be commented on far more than the first, for no good reason.

In many of the big national lotteries, a certain very large prize is shared between all people who got a winning combination. Perhaps people have a tendency to avoid "last week's" winning number. Or perhaps they have a tendency to bet last week's winning number, in the belief that it is "hot," whatever that may mean. Information of this type could increase your probability of not having to share the prize, so it could have an effect on your expected winnings.

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Assuming that the draws are fair and independent (and the evidence seems to be that they are) then it makes no difference which numbers you choose.

But if the prize pot is shared between winners then you should avoid choosing any notable pattern, including the previous draw results, as it may mean you have to share the prize with more people, lowering the expected return (or, more usually, raising the expected loss).

It only takes a small number of people to choose a notable pattern to make the proportion choosing it disproportionately large. This was shown in the Bulgarian national lotteries of 6 & 10 September 2009 when nobody chose the numbers drawn on the earlier date, but 18 did four days later when precisely the same numbers came up.

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