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I am solving an equation to find the probability that the first card of a standard, 52 card deck is a king or a heart, and I feel like I’m doing this wrong.

I set up this equation (k for king, h for heart):

$P(k$ or $h)=P(k)+P(h)$ $-(P(k) \cdot P(h))$

I first solved for $P(k)$. I knew that the denominator of $P(k)$ would be the permutation of every card in the deck:

${ }_{52} P_{52}=52 !$

I solved for the numerator by figuring out the permutations of all the cards after the king, then I multiplied by four because there are 4 kings.

${ }_{51} P_{51} \cdot 4=51 ! \cdot 4$

I then knew $P(k)=\frac{51 ! \cdot 4}{52 !}$

I already knew the denominator for $P(h)$ was $52 !$, so all I needed to do was find the numerator. It was going to be $51!$ times the amount of hearts that there were, so $13$

I then knew:

$P(h)=\frac{51 ! \cdot 13}{52 !}$

Now that I know $P(h)$ and $P(k)$, I could solve the equation:

$P(k$ or $h)=\frac{51 ! \cdot 4}{52 !}+\frac{51 ! \cdot 13}{52 !}$ $-\left(\frac{51 ! \cdot 4}{52 !} \cdot \frac{51 ! \cdot 13}{52 !}\right)$

I know that $\frac{51 !}{52 !}=\frac{1}{52}$, so I simplified down to:

$P(k$ or $h)=\frac{4}{52}+\frac{13}{52}$ $-\left(\frac{4 \cdot 13}{52^{2}}\right)$

$ =\frac{17}{52}-\frac{52}{52 \cdot 52} $

$ =\frac{17}{52}-\frac{1}{52} $

$ =\frac{16}{52} $

Did I solve this correctly? I was also wondering if there was a better way to do this.

Thank you!

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    $\begingroup$ Your answer is correct (and reduces to $\frac{4}{13}$). I think it's easier to simply count cards that are kings or hearts and then subtract the one card (the king of hearts) that's both. $\endgroup$
    – Robert Shore
    Commented Jan 26, 2022 at 1:24
  • $\begingroup$ Thank you for telling me that! I now realize that I did too much work… $\endgroup$
    – Aaron S
    Commented Jan 26, 2022 at 1:26
  • $\begingroup$ Also, you are just interested in what the top card is, not in how all the other cards are organized. So ignore the rest of the deck and just ask "what can the top card be"? There are 52 cards, and each is equally likely to be on top, so the probability of any particular card being on top is $\frac 1{52}$. There are $13$ hearts and $3$ non-heart kings, for $16$ total, so the probability will be $\frac {16}{52} = \frac{4}{13}$. No need for factorials. $\endgroup$
    – Paul Sinclair
    Commented Jan 26, 2022 at 18:07

1 Answer 1

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Community wiki answer so the question can be marked as answered:

As noted in the comments, your calculation is correct but somewhat unnecessarily complicated. You don’t need to consider the order of the remaining $51$ cards; that just multiplies both the numerator and the denominator by $51!$ and cancels out. There are $4$ kings and $13$ hearts, which would be $4+13=17$ cards, but one card is both a king and a heart and needs to be subtracted back out, so there are $16$ you’re looking for out of a total of $52$, for a probability of $\frac{16}{52}=\frac4{13}$.

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