**(Note: this problem has an error…can you find it? I will post the correction soon here, which itself has some interesting properties)**

We often make the important distinction between drawing cards with or without replacement when determining the results of card games. Clearly, if you draw a Jack, then the probability of drawing another Jack is different whether you replace and shuffle, or you leave the drawn Jack out. In an extreme case, imagine a deck with only one Jack.

After going through an example in class, and applying replacement, a student asked about the same example without replacement. I didn’t have time to go over the example in class, but I sketched how the calculation would be different, and would yield a different answer. After class, going through the calculation, I was in for a surprise.

# The problem

You draw two cards from a deck, and ask what is the probability that the first is a black card, and the second is a jack. In math notation, we want:

## With replacement

In replacement, we replace the first card after drawing it, reshuffle, and then draw the second. Thus the two events are independent.

## Without replacement

Without replacement is a little trickier to set up, because the second draw depends on the first.

**the same answer!!**

**(Note: this problem has an error…can you find it? I will post the correction soon here, which itself has some interesting properties)**

## Conclusion

The only conclusions are

- My intuition fails me sometimes, on even simple problems
- Drawing a black card on the first draw tells you no more information about drawing a jack on the second (or any other number, for that matter).

Cool!

**(Note: this problem has an error…can you find it? I will post the correction soon here, which itself has some interesting properties)**