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I have been wondering if the probability of drawing a card from a 52 card deck, obviously 1/52 probability, is different from the following.

If you get to draw a card from a 52 card deck and right after you draw the card, without seeing if it’s the correct card, you put it back in the deck. What is the probability of that drawn card being the correct card? And, after drawing the card and not looking, did you still have a 1/52 chance that the card you chose is the correct card?

More precise: For my bachelor thesis in law, I'm writing about the a Dutch doctrine in liability law, called 'Loss of a chance' where there is uncertainty about the cause of damage and therefore liability is granted as a percentage of chance that someone had right on a claim. Now, there is an article where someone claims that there is a difference between two cases. One where someone has the chance to draw a card, but is denied. He obviously lost a 1/52 chance. And, in the other case, someone has already drawn the card, but is denied knowing the answer. The author states that the loss of not knowing the outcome is not the loss of a chance, because the chance has been exercised already. I was wondering if this is mathematically endorsed, because it seems so counter-intuitive.

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    $\begingroup$ This is not clear. Whether you look at the card or not has no effect on what it is. $\endgroup$ – lulu Apr 20 at 11:01
  • $\begingroup$ Indeed, not looking at the card is like not drawing a card at all, the chances of it being the right one is still $\frac{1}{52}$. $\endgroup$ – Luyw Apr 20 at 11:12
  • $\begingroup$ Added more text to be more clear about the problem I'm working on. $\endgroup$ – NielsKuil Apr 20 at 11:12
  • $\begingroup$ Still, looking at the card has no consequences on the result. However being denied drawing a card is like a $0\%$ chance of drawing the right one since you haven't exercised your chance, or that's how I see it. $\endgroup$ – Luyw Apr 20 at 11:19
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The probability of drawing an ace of spades on your second try is still 1/52. Since you put the card back in the deck, the two drawings are independent events, hence the outcome of one does not affect the other, and certainly, whether you looked at the card or not, doesn't.

In fact, even if you looked at the card, and irrespective of what the first card is, if you put it back in, and draw another card, the probability of your second card being ace of spades is still unchanged, at 1/52.

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