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

Conditional probability , Multiplication Theorem on Probability

Conditional probability , Multiplication Theorem on Probability

- Probability is the measure of the likeliness that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty)

- We have read about Terms related to Probability,Algebra of events operations< ,Classical Probability in Previous class
- We have also read also Addition Theorems on Probability in previous classes
- Now we will learn about 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

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

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

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

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

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