PPT - Baye’s Theorem

# PPT - Baye’s Theorem - Business Mathematics and Statistics - B Com

``` 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:

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115 videos|142 docs

## FAQs on PPT - Baye’s Theorem - Business Mathematics and Statistics - B Com

 1. What is Baye's Theorem?
Baye's Theorem is a mathematical formula that calculates the probability of an event occurring based on prior knowledge of related events.
 2. How is Baye's Theorem used in real-life applications?
Baye's Theorem is widely used in various fields such as medicine, finance, and marketing. It helps in making informed decisions by updating the probability of an event based on new information.
 3. Can you explain the formula of Baye's Theorem?
Certainly! Baye's Theorem can be expressed as: P(A|B) = (P(B|A) * P(A)) / P(B), where P(A|B) is the probability of event A given that event B has occurred, P(B|A) is the probability of event B given that event A has occurred, P(A) is the probability of event A, and P(B) is the probability of event B.
 4. What are the limitations of Baye's Theorem?
Baye's Theorem assumes that the events are independent of each other, which may not always be the case in real-life situations. Additionally, it requires accurate prior probabilities, which can be challenging to obtain.
 5. Can you provide an example of how Baye's Theorem is applied in practice?
Sure! Let's say a medical test for a certain disease has a 95% accuracy rate. However, the disease is quite rare, affecting only 1% of the population. If a person tests positive for the disease, Baye's Theorem can be used to calculate the probability of actually having the disease, taking into account both the test accuracy and the prevalence of the disease.

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