Mathematics for Computer Science

16.5. Conditional Probability 547

16.5.7 Conditioning on a Single Event

The probability rules that we derived in Section 16.4.2 extend to probabilities con-
ditioned on the same event. For example, the Inclusion-Exclusion formula for two
sets holds when all probabilities are conditioned on an eventC:

Pr



A[BjC



DPr



AjC



CPr



BjC



Pr



A\BjC



:

This is easy to verify by plugging in the Definition 16.5.1 of conditional probabil-
ity.^5
It is important not to mix up events before and after the conditioning bar. For
example, the following isnota valid identity:

False Claim.

Pr



AjB[C



DPr



AjB



CPr



AjC



Pr



AjB\C



: (16.6)

A simple counter-example is to letBandCbe events over a uniform space with
most of their outcomes inA, but not overlapping. This ensures that Pr



AjB



and
Pr



AjC



are both close to 1. For example,

BWWDŒ0;9ç;

CWWDŒ10;18ç[f 0 g;

AWWDŒ1;18ç;

so

Pr



AjB



D

9

10

DPr



AjC



:

Also, since 0 is the only outcome inB\Cand 0 ...A, we have

Pr



AjB\C



D 0

So the right hand side of (16.6) is 1.8, while the left hand side is a probability which
can be at most 1 —actually, it is 18/19.

16.5.8 Discrimination Lawsuit

Several years ago there was a sex discrimination lawsuit against a famous univer-
sity. A woman math professor was denied tenure, allegedly because she was a
woman. She argued that in every one of the university’s 22 departments, the per-
centage of men candidates granted tenure was greater than the percentage of women
candidates granted tenure. This sounds very suspicious!

(^5) Problem 16.11 explains why this and similar conditional identities follow on general principles
from the corresponding unconditional identities.