## Introduction

In a previous post I made a calculation error, which arrived at an
unintuitive result - a result that still stands. I got side-tracked
with, what I thought, was an arithmetic error. I wasn't satisfied,
because my intuition still thought that the without-replacement
probability should be a smidge higher, because of the reduction in the
number of cards. Because of this, I kept thinking about the problem to
see where it went wrong. I asked another faculty member the same
question, and although I didn't receive a full answer, it was enough to
figure out that I was on the right track initially, but was just a
little sloppy. So what went wrong, and why? Let me reproduce the
problem, and the *correct* calculation this time, and then go on to see
the implications of the error.

## 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:

## The right answer, and how I know

The easiest way to be absolutely sure I had the right answer is to simply outline every possible two-hand deal, and count the number of cards in each case.

### With replacement:

```
from Game import *
deals=[]
deck=makedeck()
for card1 in deck:
deck2=makedeck()
for card2 in deck2:
deals.append([card1,card2])
found=[x for x in deals if x[0].color=='Black' and x[1].rank==11]
```

The length of "found" is 104, and the length of the "deals" is 2704 (52 x 52).

### Without replacement:

```
from Game import *
deals=[]
deck=makedeck()
for card1 in deck:
deck2=makedeck()
deck2.remove(card1) # <------ remove the card
for card2 in deck2:
deals.append([card1,card2])
found=[x for x in deals if x[0].color=='Black' and x[1].rank==11]
```

The length of "found" is 102, and the length of the "deals" is 2652 (52
x 51), which is the *same* fraction.

### With replacement

*This part was correct, and just repeated here.*

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

### Without replacement

where I have put boxes, or underline, around where I differ from the previous calculation.

### The difference

The difference comes from the term like:

In the *incorrect* version, we had

Essentially, I was being sloppy, and forgot to copy the right-hand side
of the conditional, without the *B1*.

What I find interesting, which is why I've gone to such a detail, are the following:

- how easy it is to make simple arithmetic mistakes in these sorts of problems
- how easy it is to have a subtle rewrite of a problem, and get a different answer
- how a simulation gives a lot of confidence in a result

I've found, over time, that I don't tend to trust mathematical results without a numerical result to support it.

Still, it is a cool result, and still somewhat unintuitive - at least at first. Thinking in terms of information, it makes sense - knowing that the first card is black tells you nothing about the rank of the second card.