# 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:

$P(B1,J2)$

## 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.

$\begin{array}{rcl} P(B1,J2|{\rm replace}) &=& P(B1) \times P(J2) \\ &=&\frac{26}{52} \times \frac{4}{52} = \frac{104}{2704}=0.0385 \end{array}$

## Without replacement

$\begin{array}{rcl} P(B1,J2|{\rm no-replace}) &=& P(B1) \times P(J2|B1) \\ &=&\frac{26}{52} \times \left(P(J2|B1,J1)\underline{P(J1|B1)}+P(J2|B1,\overline{J1})\underline{P(\overline{J1}|B1)}\right)\\ &=&\frac{26}{52} \times \left(\frac{3}{51} \cdot \underline{\frac{2}{26}}+\frac{4}{51} \cdot \underline{\frac{24}{26}} \right)\\ &=&\frac{26}{52} \times \underline{\left(\frac{102}{51\cdot 26} \right)}\\ &=&\frac{102}{2652}\\ &=&0.0385 \end{array}$

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

# The difference

The difference comes from the term like:

$P(J2|B1,J1)P(J1|B1)$

In the incorrect version, we had

$P(J2|B1,J1)P(J1)$

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:

1. how easy it is to make simple arithmetic mistakes in these sorts of problems
2. how easy it is to have a subtle rewrite of a problem, and get a different answer
3. 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.