Probability Examples on Bayes Theorem

In this class, We discuss Probability Examples on Bayes Theorem.

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The reader should have prior knowledge of the Bayes Theorem. Click Here.

Example 1:

Suppose five men out of a hundred and 25 women out of 10000 are color blind.

We choose a color-blind person at random.

What is the probability of the person being male?

Assume males and females are equal in number.

Solution:

P(M) = 1/2 and P(F) = 1/2

Male M and Female F are equal in number.

P(B|M) = 5/100 = 0.05

P(B|F) = 25/10000 = 0.0025

P(M|B) = (P(M)P(B|M))/(P(M)P(B|M) + P(F)P(B|F))

P(M|B) = (0.05 * 0.5)/((0.05 * 0.5) + (0.5 * 0.0025))

P(M|B) = 0.95

Example 2:

The chance that doctor A will diagnose a disease x correctly is 60%.

A patient’s chance of dying of disease x after correct diagnosis is 40%.

The chance of death after a wrong diagnosis is 70%.

A patient of doctor A died of disease x.

What is the chance of the disease being diagnosed correctly?

Solution:

Event D = Diagnoised properly

P(D) = 60/100 = 0.6

P(D’) = 1 – P(D) = 0.4.

P(Death|D) = 0.4

P(Death | D’) = 0.7

P(D|Death) = (P(D)P(Death|D))/(P(D)P(Death|D) + (P(D’)P(Death|D’))

P(D|Death) = (0.6 * 0.4)/((0.6 * 0.4) + (0.7 * 0.4))

P(D|Death) = 6/13