## Section 6.6: Bayes' Theorem

(Based on Section 6.6 inFinite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World. )

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## Bayes' Theorem

Bayes theorem gives us a way of calculating P(E|F) from a knowledge of P(F|E).

Example
E = the event that an anabolic steroid detection test gives a positive result
F = the event that the athlete uses steroids
then
P(E|F) = probability that the test is positive for an athlete who uses steroids.
P(F|E) = probability that an athlete uses steroids given that the test is positive.

A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. Your friend on the rugby team has just tested poositive. The probability that he uses steroids is:

 0.95 At most 0.95 At least 0.95 Not possible to say, based on the information

Question
What information do we need to calculate P(F|E) from a knowledge of P(E|F)?

Two additional pieces of information: we need to know P(F) (the probability that an athlete on the team is using steroids) and P(E|F') (the probability of a false positive result).

A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. What the company does not tell you is that 15% of all steroid-free individuals also test positive (the false positive rate). 10% of the rugby team members use steroids. Take

E = the event that a rugby team member tests positive
F = the event that a rugby team member uses steroids

Complete the following chart and press "Check" to check each answer:

 P(F) = P(F') = P(E|F) = P(E'|F) = P(E|F') = P(E'|F') =

Now select the correct tree from the following choices.

 Key: uses steroids does not use steroids tests positive tests negative

 B B
 B B
 B B
 B B

Here is the information again: A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. Further, 15% of all steroid-free individuals also test positive. 10% of the rugby team members use steroids. Your friend on the rugby team has just tested poositive. The probability that he uses steroids is? (Refer to the tree to calculate the answer.)

 0.413 0.8636 0.095 0.23

Notice that, in the calculation we just finished, we computed P(F|E) from the probability tree, using information such as P(E|F) and P(E|F'). If you press the "HELP" button immediately above, you will find that we used the following calculation.

Bayes' Theorem (Short Form)

If E and F are events, then

 P(F|E) = P(E|F)P(F)P(E|F)P(F) + P(E|F')P(F') .

Example

In the example we just considered,

P(E|F) = 0.95     P(E|F') = 0.15     P(F) = 0.1     P(F') = 0.9

(Press here to go back to where we calculated these probabilities.) Therefore,

P(F|E)= P(E|F)P(F) P(E|F)P(F) + P(E|F')P(F')
= (0.95)(0.1)(0.95)(0.1) + (0.15)(0.9)
0.4130,

which is the answer obtained above.

Note
There is an "extended" form of Bayes' Theorem. Consult pp. 462-463 in Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World for a discussion of the extended form of Bayes' theorem.

As accounts manager in your company, you classify 75% of your customers as "good credit" and the rest as "risky credit" depending on their credit rating. Customers in the "risky" category allow their accounts to go overdue 50% of the time on average, whereas those in the "good" category allow their accounts to become overdue only 10% of the time. What percentage of overdue accounts are held by customers in the "risky credit" category?

 20% 12.5% 93.75% 62.5%

Now try some of the questions in the true/false quiz (warning: it covers the whole of Chapter 6) by pressing the button on the sidebar, or try some of the exercises on pp. 464-467 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Last Updated: April, 1997