D. How is

Bayes theorem important in avoiding fallacies commonly made in law and

courtrooms?

Base rate fallacy

If presented with related generic, general

information and specific information (information concerning a particular case),

the mind tends to ignore the former and focus on the latter.1

This can be avoided using Bayes’ Theorem, as it takes into account prior

probability (which

is the probability assessed before making reference to certain relevant

observations, especially subjectively or on the assumption that all possible

outcomes be given the same probability) and posterior probability (which is the conditional probability that is assigned after the relevant evidence or

background is taken into account)

Example:

Drunk drivers measured by a breathalyzer.

The

conditions are as follows:

? Breathalyzers

display a false result in 5% of the cases tested.

? They never fail

to detect a person who is truly drunk.

? drivers on the road are drunk.

If a policeman stops a driver at random and the breathalyzer displays that the driver is drunk, what is the probability that the driver is truly drunk?

?

For the 1 driver who is

truly drunk, there is a 100% chance of a true

positive test result as the second condition stated. Thus, the true positive test result is 1.

?

For the other 999 drivers

who are not drunk, there is a 5% chance of false

positive test results, as the first condition stated. Thus, the false positive test results is 49.95

(999 x 5%)

From this, we can calculate

the probability that one of the drivers among the 50.95 (1 + 49.95) positive

test results, who is really drunk:

= 0.019627

This can be confirmed with

Bayes Theorem.

Notations:

D = drunk

S = sober

B = breathalyzer indicates that the

driver is drunk

Result

Drunk

Sober

Test

Positive

1

0.05

Test

Negative

0

0.95

Historic

Data

0.001

0.999

Notation form:

P(D) = 0.001

P(S) = 0.999

P(B|D) = 1.00

P(B|S) = 0.005

Figure 1: Venn Diagram depicting

probability of driver being drunk or sober and results from breathalyzer.

The probability of the

driver being drunk, given that the breathalyzer indicates that h/she is drunk

is represented by p(D|B).

According to Bayes’ theorem:

P (D|B) =

From that we can arrive at

p(D):

P(D) = P(B|D)P(D) + P(B|S) P(S)

Computing above values to

obtain value of P(D) (refer to table and following notation

form):

P(D) = (1.00 x 0.001) + (0.05 x

0.999)

= 0.05095

Computing all values into

Bayes’ Theorem formula:

P(D|B) =

= 0.019627

1 Woodcock, S. (April 5, 2017) Paradoxes of probability and other statistical strangeness