Bayes' theorem provides a way to revise prior probabilities based on new evidence and determine the probability that an effect was caused by a specific cause. Posterior probabilities refer to prior probabilities that have been updated based on new information or evidence, and are conditional probabilities. Bayes' rule is used to calculate posterior probabilities from prior probabilities and new evidence using the formula: Posterior probability of A given B = (Probability of A and B) / (Probability of B). An example is provided of using Bayes' theorem to calculate the probability that a defective item was produced by machine Y given that it was drawn randomly from a bin containing items from machines X, Y, and Z.

1. Posterior Probability
Presented by:
Bishesh K Sah
Roll No.- 14

2. Bayes’ Theorem
Developed by British mathematician Rev. Thomas
Bayes.
The procedure for revising the prior probabilities
based on new information and determining the
probability that a particular effect was due to a
specific cause.
The theorem is based on Conditional probability.

3. Posterior probability
The prior probabilities changed in the light of
new information are called Posterior or Revised
probabilities.
Posterior probabilities are always conditional
probabilities , the conditional event being the
sample information.
Thus a prior probability which is unconditional
becomes a Posterior, which is conditional by
using Bayes’ rule.

4. Continued……..
Posterior probabilities are determined with the
help of Bayes’ theorem:
P(Ai│B) = P(Ai∩B)
P(B)
Where Posterior probability of Ai given B, is the
conditional probability P(Ai│B)

5. Example:
Suppose an item is manufactured by 3 machines X, Y
& Z. All 3 machines have equal capacity & are
operated at the same rate. It is known that the
percentages of defective items produced by X, Y & Z
are 2, 7 and 12 respectively. All the items produced
by X, Y & Z are put into one bin. From this bin, one
item is drawn at random and is found to be
defective. What is the probability that the item was
produced on Y?

6. Thank You