A card is drawn from 52 cards. What is the probability that the second card is a face card? I hope the question is clear.

I believe the answer to be as follows...

Two cases -

Case 1: First card drawn is a face card. Case 2: First card drawn is not a face card.

Then the total probability is $\;\frac{12\cdot 11}{52\cdot 51} + \frac{40\cdot 12}{52\cdot 51}$.

  • 3
    $\begingroup$ No, the question isn't clear (and it belongs in the body of the post). If only one card is drawn, where does "the second card" come from? Is it drawn as well? $\endgroup$ – joriki Dec 6 '12 at 11:01
  • $\begingroup$ You are correct. $\endgroup$ – Joe Johnson 126 Dec 6 '12 at 11:09
  • $\begingroup$ @joriki The second card is from the remaining 51 cards. $\endgroup$ – adifire Dec 6 '12 at 11:29
  • $\begingroup$ What would you calculate as the probability that the 52nd card is a face card? $\endgroup$ – Hagen von Eitzen Dec 7 '12 at 21:49

Your calculation is correct. However, note that the result simplifies to $(12\cdot11+40\cdot12)/(52\cdot51)=(11+40)\cdot12/(52\cdot51)=12/52$.

This is because you can just as well ignore the first card and argue that the probability of the second card being a face card is the number of face cards, $12$, divided by the total number of cards, $52$.

  • $\begingroup$ hello , if first card is drawn ..... there remains 51 cards . If that card was not face card , it's 12/51 am i right ??? $\endgroup$ – pravin poudel Mar 20 '18 at 6:32
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    $\begingroup$ @pravinpoudel, since you understand that the probability of getting a face card drawing one card from a full deck is 12/52, let's ask this question instead: what if, before drawing that one card, you take the bottom half of the deck (26 cards) and drop them in the garbage can without looking at them? Now you only have 26 cards left. Does that change the odds of the top card being a face card? $\endgroup$ – Wildcard Jul 26 '18 at 2:42

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