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I am trying to solve a simple exercise from a probability book but I have no idea how. Given a five-card deck containing: ace of spades, king of spades, king of hearts, queen of spades, and queen of hearts, there are 2 players and each one has exactly one card. You observe evidence that one player has a picture card (king or queen). What’s the probability that the other player has a spade?

I think at conditional probabilities, but I don't know how to model this events. P(A|B) where A is the event of extracting a spade and B represents event of having a spade from a picture card. I know that if I observe picture card than the probability of being a spade is 1/2.

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closed as unclear what you're asking by JMoravitz, Ethan Bolker, Shailesh, Leucippus, Parcly Taxel May 24 '18 at 14:25

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ How many cards does each player hold? Did we see the rank of all of the one players' cards, or just that one? $\endgroup$ – C Monsour May 20 '18 at 19:26
  • $\begingroup$ It matters how the cards are split up among the players. Consider the extreme example of where the youngest player is always given all of the cards. In such a scenario, it is impossible for the older player to hold a spade (or in fact any card at all). Consider the other extreme where for each card a coin is flipped. If heads was flipped the first player gets the card. If tails, then the second player gets the card. Consider a third example where the first player is always dealt the top two cards and the second player gets the remaining three. These all have different final answers $\endgroup$ – JMoravitz May 20 '18 at 19:38
  • $\begingroup$ each player has exact one card. All that I know is that one of them has a picture card. (it can be spade or not). I have to find probability that the other player has a spade. $\endgroup$ – user1979704 May 21 '18 at 10:27
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Hint

Taking your "All that I know..." means that a random card "opened" is seen to be a face card,

either it is a spade face card (Pr = ?) in which case the other card has a Pr of $\frac14$ of being a spade,

or it is a non-spade face card (Pr = ?) in which case the other card has Pr of $\frac24$ of being a spade

Can you now frame it in terms of conditional probabilities, and apply the law of total probability ?

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  • $\begingroup$ $$Pr(A) = Pr(A|B) * P(B) + Pr(A|C) * P(C) $$ where A is the event of having a spade, B if "open" card is a spade and C if "open" card is not a spade. $$Pr(B) = \frac{1}{2}, Pr(A|B) = \frac{2}{4}, Pr(C) = \frac{1}{2}, Pr(A|C) = \frac{3}{4}$$ thus resulting $$Pr(A) = \frac{2} {8} + \frac{3}{8} = \frac{5}{8}$$. $\endgroup$ – user1979704 May 22 '18 at 12:37

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