This Chapter we would be taking a look at the below topics

Conditional probability , Multiplication Theorem on Probability

## Introduction

## What is Conditional Probability

Let E and F are two events of the random experiments. Probability of event A happening give the condition event F has happened is called Conditional probability

So conditional probability of E given F has happened is P(E | F).

Similarly, we can define P (F| E).

We can think these conditional probabilities in these ways also

P (E|F) -> Probability of occurrence of E given F has happened

Or

P(E|F) -> Probability of occurrence of E when event F is taken as sample space

**Example**

Let take an example of a random experiment of Flip a coin twice.

S = {(H, H), (H, T), (T, H), (T, T)}.

Let F be the event of first flip to be H.

And let Me be the event for two flips are not both H

Then F = {(H, H), (H, T)}, P(F) = 1/2.

E = {(H, T), (T, T), (T, H)}, P(E) = 3/4.

Now If F event occurs, in order for event E to occur, the second flip has to be T.

P(E|F) = ½

## Multiplication Theorem on Probability|Conditional probability formula

(1) Let E and F are two events of a random experiments

Then
(P (E

∩ F) = P(E) P(F|E) if P(E) ≠0

Or

P (E

∩ F) = P(F) P(E|F) if P(F) ≠0

**Proof:**
Let S be the sample space and it contains n elementary events

Let a, b, c is the number of elementary event in E, F and (E

∩ F)

Then

P(E) = a/n

P(F) = b/n

P(E

∩ F)= c/n

Now

P(E |F) =c/b

P(F |E) = c/a

Now

P(E

∩ F)= c/n

=(c/b) (b/n)

= P(F) P( E|F)
Or

P(E

∩ F)= c/n

=(c/a) (a/n)

= P(E) P( F|E)
Hence Proved

Now

P (E

∩ F) = P(E) P(F|E) if P(E) ≠0

Or

P (E

∩ F) = P(F) P(E|F) if P(F) ≠0

Can be rewritten as

P(F|E) = P (E

∩ F) / P(E)

Or

P(E|F) = P (E

∩ F) / P(F)

2) So far we have proved all about dependent events

**What are Independent Events**
Events can be "Independent", meaning each event is not affected by any other events. We can say that if the occurrence or non-occurrence of one does not affect the probability of the occurrence or non-occurrence of the other

Two events E and F are independent then

P(E|F) = P(E) and P(F|E) = P(F).

So P (E

∩ F) = P(E)P(F).

We can extend this n number of events

Also for independents events

P (E ∪ F) = P(E) + P(F) – P (E

∩ F)

P (E ∪ F) = P(E) + P(F) – P(E) P(F)

= 1 – P(E

^{c}) P(F

^{c})

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