CK-12 Probability and Statistics - Advanced

3.4. Conditional Probability http://www.ck12.org

However, we want to show a systematic way of calculating conditional probabilities. Take the ratio of the probability
of the part ofAthat falls within the reduced sample spaceB(i.e., the intersection of the two sample spacesAandB)
to the total probability of the reduced sample space.

To calculate the conditional probability that eventAoccurs, given that eventBoccurs, take the ratio of the probability
thatboth AandBoccur to the probability thatBoccurs. That is,

P(A|B) =

P(A∩B)

P(B)

For our example above, the die toss experiment, we proceed as follows:

A={observe an even number}

B={observe a number less than or equal to 3}

We use the formula,

P(A|B) =

P(A∩B)

P(B)

and get,

P(A|B) =

P(A∩B)

P(B)

=

P( 2 )

P( 1 )+P( 2 )+P( 3 )

=

1 / 6

3 / 6

=

1

3

Example:

A medical research center is conducting experiments to examine the relationship between cigarette smoking and
cancer in a particular city in the US. Let A represent an individual that smokes and letCrepresent an individual
that develops cancer. SoACrepresents an individual who smokes and develops cancer,AC′represents an individual
who smokes but does not develop cancer and so on. We have four different possibilities, simple events, and they are
shown in the table below along with their associated probabilities.

TABLE3.2:

Simple Events Probabilities

AC 0. 10

AC′ 0. 30

A′C 0. 05

A′C′ 0. 55

Figure:A table of probabilities for combinations of smoking(A)and developing cancer(C).

How can these simple events be studied, along with their associated probabilities, to examine the relationship
between smoking and cancer?

Solution:

We have