Bayes’ Theorem

4.4 Bayes’ Theorem-Continued Bayes’ theorem is used to revise previously calculated probabilities based on new information. Developed by Thomas Bayes in the eighteenth century (see references 1, 2, and 6), Bayes’ theorem is an extension of what you previously learned about conditional probability.

You can apply Bayes’ theorem to the situation in which M&R Electronics World is considering marketing a new model of televisions. In the past, 40% of the new­model televisions have been successful, and 60% have been unsuccessful. Before introducing the new model television, the marketing research department conducts an extensive study and releases a report, either favorable or unfavorable. In the past, 80% of the successful new­model television(s) had received favorable market research reports, and 30% of the unsuccessful new­model television(s) had received favorable reports. For the new model of television under consideration, the marketing research department has issued a favorable report. What is the probability that the television will be successful?

Bayes’ theorem is developed from the definition of conditional probability. To find the  conditional probability of B given A, consider Equation (4.4b) shown again below:

Bayes’ theorem is derived by substituting Equation 4.8 for P(A) in the denominator of Equation (4.4b)

P (B|A) = = P (AandB)

P (A)

P (A|B)P (B)

P (A)

Bayes’ Theorem P (Bi|A) =

P (A|B )P (B ) + P (A|B )P (B ) + ⋅ ⋅ ⋅ + P (A|B )P (B )

where B  is the ith event out of k mutually exclusive and collectively exhaustive events.

To use Equation 4.9  for the television­marketing example, let

and

P (A|Bi)P (Bi)

1 1 2 2 k k

(4.9)

i

eventS = successful television eventF = favorable report

eventS’ = unsuccessful television eventF ‘ = unfavorable report

P (S) = 0.40 P (F |S) = 0.80

P (S’) = 0.60 P (F |S’) = 0.30

Then, using equation 4.9

Table 4.4  summarizes the computation of the probabilities, and Figure 4.3  presents the decision tree. Example 4.10  applies Bayes’ theorem to a medical diagnosis problem.

Table 4.4 Bayes’ Theorem Computations for the Television­Marketing Example

Event  Prior Probability

Conditional Probability

Joint Probability  Revised Probability

successful television

0.40 0.80 0.32

unsuccessful television

0.60 0.30

P (S|F) =

=

= =

= 0.64

P(F |S)P(S)

P(F |S)P(S)+P(F |S’)P(S’)

(0.80)(0.40)

(0.80)(0.40)+(0.30)(0.60)

0.32 0.32+0.18

0.32 0.50

The probability of a successful television, given that a favorable report was received, is 0.64. Thus, the probability of an unsuccessful television, given that a favorable report was received, is  1 − 0.64 = 0.36.

Si

P (Si) P (F |Si)

P (F |Si) P (Si) P (Si|F)

S = P (S|F) = 0.32/0.50

= 0.64

S’ = 0.18

0.50

P (S’|F) = 0.18/0.50

= 0.36

Figure 4.3 Decision tree for marketing a new television

Figure 4.3 Decision tree for marketing a new television

Example 4.10 Using Bayes’ Theorem in a Medical Diagnosis Problem The probability that a person has a certain disease is 0.03. Medical diagnostic tests are available to determine whether the person actually has the disease. If the disease is actually present, the probability that the medical diagnostic test will give a positive result (indicating that the disease is present) is 0.90. If the disease is not actually present, the probability of a positive test result (indicating that the disease is present) is 0.02. Suppose that the medical diagnostic test has given a positive result (indicating that the disease is present). What is the probability that the disease is actually present? What is the probability of a positive test result?

Solution Let

and

Using Equation 4.9

event D = has disease event T = test is positive

event D’ = dose not have disease event T ‘ = test is negative

P (D) = 0.03 P (T |D) = 0.90

P (D’) = 0.97 P (T |D’) = 0.02

The probability that the disease is actually present, given that a positive result has occurred (indicating that the disease is present), is 0.582. Table 4.5  summarizes the computation of the probabilities, and Figure 4.4  presents the decision tree. The denominator in Bayes’ theorem represents P(T), the probability of a positive test result, which in this case is 0.0464, or 4.64%.

P (D|T ) =

=

= =

0.582

P(T |D)P(D)

P(T |D)P(D)+P(T |D’)P(D’)

(0.90)(0.03)

(0.90)(0.03)+(0.02)(0.97)

0.0270 0.0270+0.0194

0.0270 0.0464

Table 4.5 Bayes’ Theorem Computations for the Medical Diagnosis Problem

Event  Prior Probability

Conditional Probability

Joint Probability  Revised Probability

has disease

0.03 0.90 0.0270

does not have disease

0.97 0.02

Di

P (Di) P (T |Di)

P (T |Di) P (Di) P (Di|T )

D = P (D|T ) = 0.0270/0.0464 = 0.582

D’ = 0.0194

0.0464

P (D’|T ) = 0.0194/0.0464 = 0.418

Figure 4.4 Decision tree for a medical diagnosis problem

Problems

Learning the Basics 4.66 If   and   find

. 4.67 If   and and  find  .

P(B) = 0.05, P(A|B) = 0.80, P(B′) = 0.95, P(A|B′) = 0.40,

P (B|A)

P(B) = 0.30, P(A|B) = 0.60, P(B′) = 0.70, P(A|B′) = 0.50,

P (B|A)

Applying the Concepts

4.68 In Example 4.1, suppose that the probability that a medical diagnostic test will give a positive result if the disease is not present is reduced from 0.02 to 0.01

a.If the medical diagnostic test has given a positive result (indicating that the disease is present), what is the probability that the disease is actually present?

b. If the medical diagnostic test has given a negative result (indicating that the disease is not present), what is the probability that the disease is not present?

4.69 An advertising executive is studying television viewing habits of married men and women during prime­time hours. Based on past viewing records, the executive has determined that during prime time, husbands are watching television 60% of the time. When the husband is watching television, 40% of the time the wife is also watching. When the husband is not watching television, 30% of the time the wife is watching television.

a. Find the probability that if the wife is watching television, the husband is also watching television.

b. Find the probability that the wife is watching television during prime time.

4.70 Olive Construction Company is determining whether it should submit a bid for a new  shopping center. In the past, Olive’s main competitor, Base Construction Company, has  submitted bids 70% of the time. If Base Construction Company does not bid on a job, the  probability that Olive Construction Company will get the job is 0.50. If Base Construction  Company bids on a job, the probability that Olive Construction Company will get the job is  0.25.

a. If Olive Construction Company gets the job, what is the probability that Base Construction Company did not bid?

b. What is the probability that Olive Construction Company will get the job?