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I am playing a card game. The goal is to withdraw randomly a card until there is no more. The symbol of the last card is the winning one, ie: if the last card is a spike, spike wins. Right now there are already 10 heart drawn, 10 spikes, 11 clubs and 12 diamonds; What is the probability that clubs win ? The deck of cards has a total of 52 cards, 13 of each symbol. Or how to solve the problem i a general way ? Let's say n heart, m spikes, k clubs, i diamonds already drawn. Thanks for your help.

Pistolero81
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  • Consider the case of the deck having originated with the appropriate number of "remaining" cards of each suit. From there, recall that the last card in the deck is equally likely to be any of the remaining cards in exactly the same way that the first card left in the deck is equally likely to be any of the remaining cards. – JMoravitz Mar 20 '24 at 15:47
  • Thanks for your comment ! I dont really understand how it can be equal. If for instance there is 10 clubs remaining and only 2 spikes it is more likely that the last card drawn will be a club. – Pistolero81 Mar 21 '24 at 18:48
  • If there are if there are ten clubs remaining and two spades and no other cards left in the deck... i.e. you have a deck with 12 cards remaining... the probability that the next card is a club will be $\dfrac{10}{12}$. Similarly... the last card is also going to be a club with probability $\dfrac{10}{12}$ in exactly the same way as the calculation goes for the next card. The last card was equally likely to be any of those twelve cards. The next card was equally likely to be any of those twelve cards. – JMoravitz Mar 21 '24 at 23:46
  • Do not confuse yourself by thinking "well, what if the next card drawn was a club... that changes things" Sure... it would, if we cared at all about keeping track of what the next card was and conditioned appropriately on that. We don't need to deal with hypotheticals, we don't need to deal with conditional probability... we just go straight to the answer. – JMoravitz Mar 21 '24 at 23:48
  • Ok get it ! Thanks for the explanation – Pistolero81 Mar 24 '24 at 15:01

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