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I'm answering the following question: from a standard deck of 52 cards, in 5 draws, given that you've already drawn a king, what is the probability of drawing at least 1 more king?

The way I thought about it is to just focus on the 4 remaining cards you can draw. The probability of drawing at least 1 more king (and therefore having a total of 2) is $1-$ the probability that no kings are drawn in the next 4.

Since there are 51 cards left in the deck and there are 3 kings left, this would be $1-(\frac{48}{51}\cdot \frac{47}{50} \cdot \frac{46}{49}\cdot \frac{45}{48})$ which is about 0.22.

So given that you've drawn one king, the probability of drawing at least one more in the remaining draws is 0.22.

First I just want to make sure this is correct, and second, how would I use a more typical "conditional probability" approach?

fmtcs
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  • Clarify... are you drawing a single card... looking at it and seeing that it is a king... and then you draw four more cards, wondering to yourself whether there might be an additional king among those four? Or are you drawing five cards, passing them to a friend and asking your friend "is there at least one king in the hand?", they reply yes, and then you wonder to yourself whether there are in fact at least two kings? – JMoravitz Sep 23 '21 at 18:01
  • Those are both very different questions... similar to the difference in the Boy-Girl problem... "Given a family has two children and the elder child is a girl, what is the probability both are girls" to which the answer is $\frac{1}{2}$ versus "Given a family has two children and at least one is a girl, what is the probability both are girls" to which the answer is $\frac{1}{3}$ – JMoravitz Sep 23 '21 at 18:02
  • @JMoravitz You know that the first card you draw is a king, and you want to know what the probability of getting at least one more king in the four remaining draws – fmtcs Sep 23 '21 at 18:12
  • In that case, your answer is correct... however I will emphasize that this is not the answer to the common interpretation of the question in your question title. – JMoravitz Sep 23 '21 at 18:12
  • @JMoravitz I actually don't quite see the distinction...if your friend tells you there's 1 king, wouldn't you go through the same process to figure out what the probability of there being another king is? – fmtcs Sep 23 '21 at 18:13
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    The distinction? In the case of your friend telling you that at least one of the cards among your five is a king he isn't telling you which one it is. It could just as well have been the second card or fifth card or third card that he is referring to. Your calculations are for if it were very specifically always the first card that is being referred to and no other which misses out on cases where there are two kings but neither were the first card. Similarly, it misses out on cases where there were one king but it did not occur in first position. – JMoravitz Sep 23 '21 at 18:15
  • @JMoravitz but if he tells you there's at least 1 king, can't you just consider that the first one you draw and then proceed in the same way? – fmtcs Sep 23 '21 at 18:17
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    Absolutely not. Again... see the boy-girl problem. The answer to the problem of "given that among the five cards there is at least one king in any position, what is the probability that there are at least two kings"... $\Pr(X\geq 2\mid X\geq 1) = \dfrac{\Pr(X\geq 2)}{\Pr(X\geq 1)} = \dfrac{\binom{52}{5}-\binom{4}{1}\binom{48}{4}-\binom{48}{5}}{\binom{52}{5}-\binom{48}{5}}\approx 0.122$. It is critical that you understand the difference in these problems. – JMoravitz Sep 23 '21 at 18:19
  • See also the two daughter problem and boy born on a tuesday - is it just a language trick? – JMoravitz Sep 23 '21 at 18:25
  • It's making sense now. So out of all the possibilities for drawing 5 cards from a deck, a certain number of them include at least 2 kings. Of those, some had a king drawn first and others didn't. So if you know the first card is a king, you only get those possibilities where the first card was a king, but if you only know 1 of them is a king, you get a larger number of possibilities. To me that makes sense, because knowing the first card is a king narrows down the possibilities (all those where 2 or more kings are drawn). Thanks for pointing something out that never even crossed my mind – fmtcs Sep 23 '21 at 18:44

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