PPT - Bayes Theorem | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year PDF Download

 Page 1


Bayes' Rule
Page 2


Bayes' Rule
Bayes' rule
? True Bayesians actually consider conditional probabilities as
more basic than joint probabilities. It is easy to define
P(A|B) without reference to the joint probability P(A,B). To
see this note that we can rearrange the conditional
probability formula to get:
P(A|B) P(B) = P(A,B)
Page 3


Bayes' Rule
Bayes' rule
? True Bayesians actually consider conditional probabilities as
more basic than joint probabilities. It is easy to define
P(A|B) without reference to the joint probability P(A,B). To
see this note that we can rearrange the conditional
probability formula to get:
P(A|B) P(B) = P(A,B)
Bayes' Rule
But by symmetry we can also get:
P(B|A) P(A) = P(A,B)
It follows that:
Or, 
Which is the so-called Bayes' Rule.
Page 4


Bayes' Rule
Bayes' rule
? True Bayesians actually consider conditional probabilities as
more basic than joint probabilities. It is easy to define
P(A|B) without reference to the joint probability P(A,B). To
see this note that we can rearrange the conditional
probability formula to get:
P(A|B) P(B) = P(A,B)
Bayes' Rule
But by symmetry we can also get:
P(B|A) P(A) = P(A,B)
It follows that:
Or, 
Which is the so-called Bayes' Rule.
Bayes Rule Example
? Suppose that we have two bags each containing black and white balls. 
One bag contains three times as many white balls as blacks. The other 
bag contains three times as many black balls as white. Suppose we 
choose one of these bags at random. For this bag we select five balls at 
random, replacing each ball after it has been selected. The result is that 
we find 4 white balls and one black. What is the probability that we 
were using the bag with mainly white balls?
Page 5


Bayes' Rule
Bayes' rule
? True Bayesians actually consider conditional probabilities as
more basic than joint probabilities. It is easy to define
P(A|B) without reference to the joint probability P(A,B). To
see this note that we can rearrange the conditional
probability formula to get:
P(A|B) P(B) = P(A,B)
Bayes' Rule
But by symmetry we can also get:
P(B|A) P(A) = P(A,B)
It follows that:
Or, 
Which is the so-called Bayes' Rule.
Bayes Rule Example
? Suppose that we have two bags each containing black and white balls. 
One bag contains three times as many white balls as blacks. The other 
bag contains three times as many black balls as white. Suppose we 
choose one of these bags at random. For this bag we select five balls at 
random, replacing each ball after it has been selected. The result is that 
we find 4 white balls and one black. What is the probability that we 
were using the bag with mainly white balls?
Solution
Let A be the random variable "bag chosen" then A={a1,a2} where a1 
represents "bag with mostly white balls" and a2 represents "bag with mostly 
black balls" . We know that P(a1)=P(a2)=1/2 since we choose the bag at 
random. 
Let B be the event "4 white balls and one black ball chosen from 5 
selections". 
Then we have to calculate P(a1|B). From Bayes' rule this is: