Introductory Biostatistics

marginal probabilities above, calculated separately forXandY, are summa-
rized and displayed in Table 3.2. Observe that the four cell probabilities add
to unity [i.e., one of the four eventsðX¼þ;Y¼þÞorðX¼þ;Y¼
Þor
ðX¼
;Y¼þÞor ðX¼
;Y¼
Þis certain to be true for a randomly
selected individual from the population]. Also note that the joint probabilities
in each row (or column) add up to themarginalorunivariate probabilityat the
margin of that row (or column). For example,

PrðX¼þ;Y¼þÞþPrðX¼
;Y¼þÞ¼PrðY¼þÞ

¼ 0 : 015

We now consider a third type of probability. For example, thesensitivityis
expressible as

sensitivity¼

154

379

¼ 0 : 406

calculated for the eventðX¼þÞusing the subpopulation havingðY¼þÞ.
That is, of the total number of 379 persons with cancer, the proportion
with a positive test result, is 0.406 or 40.6%. This number, denoted by
PrðX¼þjY¼þÞ, is called aconditional probability(Y¼þbeing the condi-
tion) and is related to the other two types of probability:

PrðX¼þjY¼þÞ¼

PrðX¼þ;Y¼þÞ

PrðY¼þÞ

or

PrðX¼þ;Y¼þÞ¼PrðX¼þjY¼þÞPrðY¼þÞ

Clearly, we want to distinguish this conditional probability,
PrðX¼þjY¼þÞ, from the marginal probability,PrðX¼þÞ. If they are
equal,

PrðX¼þjY¼þÞ¼PrðX¼þÞ

TABLE 3.2

X

Y þ
Total

þ 0.006 0.009 0.015

0.015 0.970 0.985

Total 0.021 0.979 1.00

PROBABILITY 113