Mathematics for Computer Science

Chapter 17 Conditional Probability720

everyone with markerEhas markerA,

everyone with markerAhas markerB,

everyone with markerBhas markerC, and

everyone with markerChas markerD.

In such a scenario, the probability of a match is

PrŒEçD

1

170

:

So a stronger independence assumption leads to a smaller bound on the prob-
ability of a match. The trick is to figure out what independence assumption is
reasonable. Assuming that the markers aremutuallyindependent may wellnotbe
reasonable unless you have examined hundreds of millions of blood samples. Oth-
erwise, how would you know that markerDdoes not show up more frequently
whenever the other four markers are simultaneously present?

Problems for Section 17.4

Homework Problems

Problem 17.1.
The Conditional Probability Product Rule fornEvents is

Rule.

PrŒE 1 \E 2 \:::\EnçDPrŒE 1 ç
Pr



E 2 jE 1



Pr



E 3 jE 1 \E 2



Pr



EnjE 1 \E 2 \:::\En 1



:

(a)Restate the Rule without using elipses (... ).

(b)Prove it by induction.

Problems for Section 17.5

Practice Problems

Problem 17.2.
Dirty Harry places two bullets in random chambers of the six-bullet cylinder of his
revolver. He gives the cylinder a random spin and says “Feeling lucky?” as he
holds the gun against your heart.