Page 38 - Probability and Statistical Inference

1. Nations of Probability 15

It is clear that P(A ) = 1/2 for i = 1, 2, whereas we have P(B | A ) = 12/20 and
1
i
P(B | A ) = 8/18. Now, applying the Bayes Theorem, we have
2

which simplifies to 20/47. Thus, the chance that the randomly drawn blue marble
came from the urn #2 was 20/47 which is equivalent to saying that the chance
of the blue marble coming from the urn #1 was 27/47.
The Bayes Theorem helps in finding the conditional
probabilities when the original conditioning events
A , ..., A and the event B reverse their roles.
1 k
Example 1.4.7 This example has more practical flavor. Suppose that 40%
of the individuals in a population have some disease. The diagnosis of the pres-
ence or absence of this disease in an individual is reached by performing a type
of blood test. But, like many other clinical tests, this particular test is not per-
fect. The manufacturer of the blood-test-kit made the accompanying informa-
tion available to the clinics. If an individual has the disease, the test indicates
the absence (false negative) of the disease 10% of the time whereas if an indi-
vidual does not have the disease, the test indicates the presence (false positive)
of the disease 20% of the time. Now, from this population an individual is
selected at random and his blood is tested. The health professional is informed
that the test indicated the presence of the particular disease. What is the prob-
ability that this individual does indeed have the disease? Let us first formulate
the problem. Define the events

A : The individual has the disease
1
c
A : The individual does not have the disease
1
B: The blood test indicates the presence of the disease
Suppose that we are given the following information: P (A ) = .4, P (A ) = .6,
c
1
1
c
P(B | A ) = .9, and P(B | A ) = .2. We are asked to calculate the conditional
1
1
c
probability of A given B. We denote A = A and use the Bayes Theorem. We
1
1
2
have
which is 3/4. Thus there is 75% chance that the tested individual has the dis-
ease if we know that the blood test had indicated so. !