Conditional Probabilities and Independence
The following is excerpted from an article written by Paul Recer, Associated Press, that appeared
in the February 6, 2002 Des Moines Register.
In a study in today’s Journal of the National Cancer Institute, researchers found that
people who used tanning devices were more likely to have common kinds of skin cancer
than were people who did not use the devices.
Researchers looked at the medical records and interviewed 1436 people. Of these, 896
had basil cell or squamous cell skin cancer. The other 540 people did not have either
cancer. Among the skin cancer patients, 190 reported using tanning devices at some
time. Of those without skin cancer, only 75 had used tanning devices.
Have skin cancer?
Use tanning devices?
Yes
No
Yes
190
706
896
No
75
465
540
265
1171
1436
If a person is selected at random from this group of 1436 people, what is the probability that …
a) The person has skin cancer?
P(skin cancer) =
896
= 0.624
1436
b) The person has skin cancer given they use tanning devices?
P(skin cancer | tanning) =
190
= 0.717
265
c) The person uses tanning devices?
P(tanning) =
265
= 0.185
1436
d) The person uses tanning devices given they have skin cancer?
P(tanning | skin cancer) =
190
= 0.212
896
e) Are the events skin cancer and using tanning devices independent? Justify your answer.
No, because the unconditional probability of having skin cancer is not the same as
the conditional probability of having skin cancer given use of tanning devices. No,
because the unconditional probability of using tanning devices is different from the
conditional probability of using tanning devices given skin cancer.
P(skin cancer) = 0.624 ≠ 0.717 = P(skin cancer | tanning)
P(tanning) = 0.185 ≠ 0.212 = P(tanning | skin cancer)