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Let's say we have a 26 card deck(13 hearts and 13 spades), and we want to find the probability that all hearts are drawn before all spades. What is this value? What if we have 52 cards?

For the first case, the way I thought about it was the following. If we have 26 cards, we can arrange them n 26! ways, and if we wanted all the hearts to come before all the spades, this can be done in 13!13! ways since there are 13! ways to arrange each suit of 13 cards. But the answer does not match the probability of 1/2 that is given in the book. I get, 13!13!/26!

The book also mentions that the second probability is also the same. This, I don't understand at all. If someone could please explain to me the answer to this question, as well as why my approach doesn't work, I would be grateful! Thank you so much!

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They’re not asking for the probability that all of the hearts are drawn before any spades are drawn: they’re asking for the probability that all of the hearts are drawn before the spades have been completely exhausted, i.e., before the last spade is drawn. Do you see now why the probability is $\frac12$ regardless of how many other cards are in the deck?

Added: Suppose that you have deck of $n$ cards that consists of $13$ hearts, $13$ spades, and $n-26$ other cards that are neither hearts nor spades. Let $H$ be the set of permutations of the deck in which the last heart comes before the last spade, and let $S$ be the set of permutations of the deck in which the last spade comes before the last heart. Every permutation of the deck is in exactly one of the sets $H$ and $S$, and we want to know the probability that a randomly chosen permutation is in $H$.

Let $p$ be a permutation in $H$. Go through the deck and interchange the first heart with the first spade, the second heart with the second spade, and so on, until you’ve completely interchanged the hearts and spades. Call the new permutation $p'$; it will be in $S$. And if you perform the same operation on $p'$, you’ll get $p$ back. In other words, we can pair each permutation $p$ in $H$ with a unique permutation $p'$ in $S$: we have a bijection between $H$ and $S$. $H$ and $S$ must therefore be the same size, so exactly half of the $n!$ permutations of the deck are in $H$, and the probability that a randomly chosen one is in $H$ must be $\frac12$.

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  • $\begingroup$ I understand that I interpreted the question wrong, but can you help explain the answer? and why is it the same for a 52 deck card? $\endgroup$
    – user832014
    Commented Oct 5, 2020 at 20:40
  • $\begingroup$ @user00000000001899: Suppose that you have $n$ cards, $13$ of them hearts, $13$ of them spades, and the rest something else. There are $n!$ possible permutations of the deck. In some of them the last heart comes before the last spade, and in some the last spade comes before the last heart. Can you see why there are exactly as many permutations of the first kind as there are of the second? $\endgroup$ Commented Oct 5, 2020 at 20:42
  • $\begingroup$ So you're saying either the last card is a spade, or the last card is a heart, and so the probability is 1/2? $\endgroup$
    – user832014
    Commented Oct 5, 2020 at 20:44
  • $\begingroup$ @user00000000001899: No, the last card of the deck might be neither a heart not a spade. But if you were to throw away all of the cards that aren’t hearts or spades, then yes, the probability is $\frac12$ that the last card of that reduced deck is a heart. At this point I think that I should probably expand my answer to explain exactly why that is true. $\endgroup$ Commented Oct 5, 2020 at 20:47
  • $\begingroup$ I think so too! $\endgroup$
    – user832014
    Commented Oct 5, 2020 at 21:25
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Your answer of $\frac {13!13!}{26!}$ is correct, as is the reasoning you used to get there. I don't know where $\frac 12$ comes from. Probably they were answering a different question.

The simple answer to why the $52$ card deck probability is the same is that you can ignore all the other cards. Deal out the whole deck. Removing all the clubs and diamonds will not change whether the hearts come before the spades.

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All that matters in the 26 card deck is the last card. what are the odds that the last card is a heart? in a 52 card deck, all the diamonds and clubs may as well be blank. They can be ignored and once again all that matters is what the last spade or heart drawn is. What are the odds that the last spade or heart drawn is a heart?

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