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My question will be similar to the one asked here Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse?

I understand that each lottery is an independent event so selecting any numbers each time give you the same probabilities of winning, I assume that the game is fair.

If we think about infinite monkey theorem for a while, we can get to the conclusion that in the infinite time each combination in the lottery will be drawn at some point with probability one.

Then this might be a justification for someone to always use the same numbers for each lottery. Of course any collection of generations is always finite, and this "justification" is an abstraction.

My question is assuming that we switch numbers at random for the each lottery, do we have a probability one that we will win in the infinite time?

Are there any assumptions, or different points of view that might justify that choosing the same combination is time can be better.

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Teresa
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  • Welcome to math stack exchange! Assuming the independence of the drawn numbers it does not matter whether a player changes the numbers every week or plays the same numbers every week. In both cases, a win would come with probability $1$ at some point. – Peter Nov 13 '15 at 23:12
  • I really think it is the same question. Why do you think it isn't the same? – hardmath Nov 13 '15 at 23:12
  • If you don't mind an approximate answer, the probability of winning the lottery is zero. – Gregory Grant Nov 13 '15 at 23:16
  • Thank you for your comments. I understand now, say if $p$ is a probability of winning in the lottery, then it is the same regardless the combination that we choose. Now in the infinite time we will have the probability $\sum_{k=0}^{n} {n \choose k}p^k (1-p)^{n-k} = (1+1-p)^{n} = 1^n$ which tends to $1$ as $n$ to $\infty$, regardless keeping or switching the numbers. – Teresa Nov 14 '15 at 00:06

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