Mathematics for Computer Science

17.2. Independence 575

LikewiseŒMD1çis the eventfT T TCHHHgand has probability1=4.

More generally, any assertion about the values of random variables defines an

event. For example, the assertion thatC 1 defines

ŒC1çDfT T T;T TH;THT;HT Tg;

and so PrŒC1çD1=2.

Another example is the assertion thatC
Mis an odd number. This is an obscure

way of saying that all three coins came up heads, namely,

ŒC
Mis oddçDfHHHg:

Think about it!

17.2 Independence

The notion of independence carries over from events to random variables as well.

Random variablesR 1 andR 2 areindependentiff for allx 1 ;x 2 , the two events

ŒR 1 Dx 1 ç and ŒR 2 Dx 2 ç

are independent.

For example, areCandMindependent? Intuitively, the answer should be “no.”

The number of heads,C, completely determines whether all three coins match; that

is, whetherM D 1. But, to verify this intuition, we must find somex 1 ;x 22 R

such that:

PrŒCDx 1 ANDMDx 2 ç¤PrŒCDx 1 ç
PrŒMDx 2 ç:

One appropriate choice of values isx 1 D 2 andx 2 D 1. In this case, we have:

PrŒCD 2 ANDMD1çD 0 ¤

1

4

3

8

DPrŒMD1ç
PrŒCD2ç:

The first probability is zero because we never have exactly two heads (C D 2 )

when all three coins match (M D 1 ). The other two probabilities were computed

earlier.

On the other hand, letH 1 be the indicator variable for event that the first flip is a

Head, so

ŒH 1 D1çDfHHH;HTH;HHT;HT Tg: