http://www.ck12.org Chapter 3. An Introduction to Probability
A:{individual smokes}
C:{individual develops cancer}
A’:{individual does not smoke}
C’:{individual does not develop cancer}A very powerful way of determining the relationship between cigarette smoking and cancer is to compare the
conditional probability that an individual gets cancer, given that he/she smokes with the conditional probability
that an individual gets cancer, given that he/she does not smoke. In other words, we want to compareP(C|A)with
P(C|A′):
P(C|A) =
P(A∩C)
P(A)
Before we enter our data into the formula, we need to calculate the value of the denominator.P(A)is the probability
of the individuals who smoke in the city under consideration. To calculate it, remember that the probability of an
event is the sum of the probabilities of all its simple events. Thus
P(A) =P(AC)+P(AC′)
= 0. 10 + 0. 30
= 0. 40
=40%
This tells us that according to this study, the probability of finding a smoker, selected at random from the sample
space (the city), is 40%. Continuing on with our calculations,
P(C|A) =
P(A∩C)
P(A)
=
P(AC)
P(A)
=
0. 10
0. 40
= 0. 25 =25%
Similarly, we calculate the conditional probability of a nonsmoker that develops cancer:
P(C|A′) =
P(A′∩C)
P(A′)
=
P(A′C)
P(A′)
=
0. 05
0. 60
= 0. 08 =8%
WhereP(A′) =P(A′C) +P(A′C′) = 0. 05 + 0. 55 = 0. 6 =60%. It is also equivalent to using the complementary
relationP(A′) = 1 −P(A) = 1 − 0. 40 = 0. 60.
So what is our conclusion from these calculations? We can clearly see that there exists a relationship between
smoking and cancer: The probability that a smoker develops cancer is 25% and the probability that a nonsmoker
develops cancer is only 8%. Taking the ratio between the two probabilities, 25%÷8%= 3. 125 ,which means a
smoker is more than three times more likely to develop cancer than a nonsmoker. Keep in mind, however, that it
would not be accurate to say that smoking causes cancer but it does suggest a strong link between smoking and
cancer.