I calculated that the probability of drawing a king and then a heart from a deck of cards is
$\frac{1(12) + 3(13)}{\text{Permutation}(52,2)}=\frac{1}{52}$
However, I also noticed that this is the same as the probability of drawing the king of hearts, which is also $\frac{1}{52}$
Is this just a coincidence or is there a reason why this happens?
Best Answer
One possible interpretation of your question is as follows. Let $A$ be the event the first draw results in a King, and $B$ be the event the second draw results in a heart. Why are the events $A$ and $B$ independent?
That they are independent can be verified by the cases computation in the post above. But let us see why the result is intuitively clear.
The probability of a heart on the second draw, is, like the probability of a heart of the first draw, or the seventeenth, equal to $\frac{1}{4}$.
Now suppose that we are told the first card drawn was a King. Should that change our estimate of the probability that the second is a heart? If so, being told the first draw was a Jack should change our estimate in exactly the same way, as should being told that the first draw was a $9$.
Since all ranks are equally likely to be drawn first, the conditional probability that the second is a heart given the first is of a specified rank is the same as the plain unconditional probability that the second card is a heart.
This says that $A$ and $B$ are independent, that is, $\Pr(A\cap B)=\frac{1}{13}\cdot\frac{1}{4}$.