Question #3002

Dear sir/madam
I wonder if you can help... I must first say how difficult it has been to find a website for someone to hopefully answer my PROBABILITY question, but somehow iv'e been drawn to you !
For nearly a year iv'e been going to a local pub quiz[wales, u.k.] ..after the quiz a raffle is drawn and the winner of the ticket [which is around £55] has to guess a playing card [QUEEN OF HEARTS]to win a £1,000. Here's how it works..on the bar wall in the pub there's a glass cabinet just big enough to contain 52 playing cards - all the cards have their backs showing and secured with a drawing pin[i think that's a thumbtack in the States] and its all under lock and key. in fact, last week i won the raffle myself but guessed the wrong card-so no £1,000 !
But recently i became aware of this card raffle as a very intriguing PROBABILITY puzzle, so along with my friend who's in my team, we did 200 simulations between us and the result was fairly even..QH came up EVERYWHERE as one would expect. However, iv'e become aware of what seems like a gaping BIG PARADOX which I can't get to grips with..here it is: from the onset the probability of picking QH[in this case] is 51-1 against. The next go it's 50-1 against,then 49-1, etc. etc. Now, when you get halfway through the pack to the 26th card the probability is still a whopping 25-1 against you picking the QH... do you agree? But the PARADOX is this..if you have turned over 26 cards which is half of the 52 cards, then YOU MUST HAVE AN EQUAL CHANCE OF THE QH TO BE IN YOUR 26 CARD PILE!
This is what I dont get..everything is stacked against you from the first card at 51-1 against right up to the 26th card which is 25-1 against ........ yet, when you get to that stage everything tells you thet you have a 50-50 chance of the QH being in your first 26 card choice !!! How can something go from 25-1 to 50-50 in an instant ?
I know there's a discrepency somewhere but for the life of me I just cannot see it...in simple terms[as i'm no expert on probabilities] could you help me to understand what's going on here.
Thankyou
Yours Sincerely
Wyn

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