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# Let’s begin today’s session again with a fun fact about probability and statistics:

“Flipping a coin repeatedly rarely comes out evenly.

In 100 flips, there’s less than 8% chance that you’ll have 50 Heads and 50 Tails.”

Amazing isn’t it?

Let’s focus on what we will learn today. So, the objective for today is to continue what we have left yesterday:

1. Where does the conditional probability formula come from?

2. Bayes Theorem

Let’s begin now,

1. Where does the Conditional Probability formula come from?

The formula for conditional probability is derived from the probability multiplication rule, P(A and B) = P(A)*P(B|A).

Step by step, here’s how to derive the conditional probability equation from the multiplication rule:

Step 1: Write down the multiplication rule:

P (A and B) = P(A)*P(B|A)

Step 2: Divide both sides of the equation by P(A):

P (A and B)/ P(A) = P(A)*P(B|A)// P(A)

Step 3: Cancel P(A)on the right side of the equation:

P (A and B)/P(A) = P(B|A)

Step 4: Rewrite the equation:

P(B|A) = P (A and B)/P(A)

2. Bayes theorem

Bayes theorem is a formula that describes how to update the probabilities of hypothesis when given evidence. It is based on the conditional probability but can be used to powerfully reason about a wide range of problems involving belief updates. The formula for Bayes theorem is: P(A|B) = P(A)*P(B|A)/P(B)

Where,

P(A|B) is how often A happens given that B happens.

P(B|A) is how often B happens given that A happens.

P(A) is how likely A is on its own

And P(B) is how likely B is on its own

Let’s discuss an Example to learn the mathematical purpose of Bayes Theorem:

There are 100 people at a party, and we are examining how many wear blacks or not, and if a man or not, and get these numbers:

The probability of being a man is P(Man) = 40/100

The probability of wearing black is P(Black) = 25/100

The probability that a man wears black is P(Black|Man) = 5/40

So, here we have to find the probability that a person wearing black is a man i.e. P(Man|Black)

P(Man|Black) = P(Man)*P(Black|Man)/P(Black)

=0.4*0.125/0.25

=0.2

Please note that it is a very basic example of Bayes theorem, it is used for many complicated tasks and given tremendous results with accuracy.

Let’s test what you’ve learned :-

Written by Vivek Mittal ( Graduate in B.com (hons) from University of Delhi and 2 Actuarial Papers passed from IFOA )

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