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Hello everyone : here's Jack's little story

So let's say the lotto Jack's playing is: $5$ numbers between $1$ and $49$

And let's say he likes to play : $ 28 , 13 , 35 , 41 , 8 $

I know that to win we have : $49 \times 48 \times 47 \times 46 \times 45 $ and we have $5!$ permutation ...

Boom , he WON .

Jack says that he will not play with this combination anymore because he already won with it.

It's like winning with $1,2,3,4,5$ or $2,4,6,8.10$ , or also $5,10,15,20,25$ Jack says that we may have more chance to win with this combination because we never saw a draw like that.

I don't think he's right but I see what he meant by that

Are all these statements true?

RobPratt
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    if lotto numbers are uniformly distributed and drawings do not depend on previous results, it's just as likely to win twice with the same combination as with any other. – Vasili Dec 02 '20 at 19:52
  • Alright , it's a little bit disturbing, it feels like it's rarer to win . Thanks for your comment –  Dec 02 '20 at 19:54
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    Think about throwing a die, the probability to get any particular number is $1 \over 6$. Now let's throw it the second time and the probability to get any particular number is $1 \over 6$ again. If order of numbers is important similar to lotto's combinations, any combination of two numbers has equal probability of $1 \over 36$. – Vasili Dec 02 '20 at 20:04
  • Does this answer your question? Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse? –  May 21 '23 at 20:58

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Both of Jack's assertions are examples of the so-called gambler's fallacy, that an event which has happened more often in the past is less likely to happen in the future, or one which has happened less often in the past is more likely to happen in the future. This is a fallacy whenever (as with lotto draws) the event in question has the same probability of occurring regardless of what has happened in the past. For lotto draws which choose $5$ numbers out of $49$ without replacement there are $\ {49\choose5}=$$\frac{49!}{5!\,44!}\ $ equally likely possible combinations of $5$ numbers that can be drawn, so, if you have just one entry, your probability of winning is always $\ \frac{5!\,44!}{49!}\approx5.2\times10^{-7}\ $, regardless of what numbers you choose, or what numbers have ever been drawn in the past.

lonza leggiera
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