Probability

Question

Please explain how $P(A ∩ B) = P(A)P(B)$?

 If $A$ and $B$ are independent.

Comments

That's the definition of two events being independent - there's no proof or why.

Answer 1

Independent Event : two Events are said to be independent if knowledge of one event happening doesn't effect the probability of other event happening .

let us say ,

we have a fair coin

P(H) = 1/2

P(T) = 1/2

if i toss it two times what's the probability of  two heads ?? (H H)

Now , I actually think of it's happening of two event

ist event is getting a head in the ist flip (H1)

2nd even is getting a head in the 2nd flip (H2)

Now , I want to find the intersection of this two

P(H1 $\cap$ H2 )  = probability of getting a head in ist flip * probability of getting a head  in the 2nd flip given that ist flip is already occurs

P(H1 $\cap$ H2 ) = P(H1) * P(H2/H1)      [ from multiplication theorem ]

                                 = 1/2 * 1/2  

what i mean to say is  the probability of getting a head in the 2nd flip  given that  already  head has occur  in ist flip , it didn't change the P(H2/H1) = 1/2

so , don't confuse between mutual exclusive and independent even

 

therefore ,  if you  understand this in terms of conditional probability formula

given A and B are two independent event

P(A/B) = P(A) ---- > i

P(B/A) = P(B) ----> ii

P(A $\cap$ B ) = P(A) * P(B/A)

                            = P(A) * P(B)  -----> from (i)

Comments

Thanks

Answer 2

Consider Conditional Probability for A and B

P(A ∩ B)=P(A)*P(B/A)--->here P(B/A) means what is the probability that B will happen given that A has already happened.

As A has already happened and if WE ARE NOT considering any replacement P(A)=n(A)/n(S) means that the sample space has changed.So it will definitely impact the P(B) as P(B)=n(B)/n(S) (as n(S) has changed).

But when we consider that the replacement is being done,meaning that the sample space will not going to change even after A happens i.e. n(S) will remain the same. then P(B/A) is nothing but P(B) only as whatever damage was done by A is not reflected on B.

Which means A and B are independent.

So P(A ∩ B)=P(A)*P(B)

 

Answer 3

$P(\frac{event\: A}{event\: B})=\frac{P(event\: A\: \cap \: event\: B)}{P(B)}$

$\Rightarrow$${P(event\: A\: \cap \: event\: B)}=P(\frac{event\: A}{event\: B})*P(B)$

But we know that event A and B are independent. Therefore, the probability of event A = P(A), whether event B happens or not.

i.e. $P(\frac{event\: A}{event\: B}) =P(\frac{event\: A}{NOT \: event\: B})=P(event \, A)$