Question

If A is a subset of B, prove that P(B|A)=1

Answer 1

probability

Answer 2

@SKMC It's given that A is a subset of B
so all the elements of A are also in B
we need to find the probability of occurrence of B given that A has occurred, now A has already occurred and also A is a subset of B, so B would have occurred too, so probability will always be 1

Answer 3

@SKMC did you understand?

Answer 4

I can explain you using an example, do you want me to do so?

Answer 5

I understand the concept, I am just trying to prove it in the correct notation
So far I have the following...

Answer 6

\[P(B|A) = P(AB)P(B)/P(A)\]

Answer 7

So i need to get to \[P(A) + P(B) =1\]

Answer 8

Oh, your approach is right but
\[P(B|A)=P(A\cap B)/P(B)\]
What is \(P(A\cap B )\)?

Answer 9

@SKMC A is a subset of B, so what's \(P(A\cap B)\)?

Answer 10

A & B simultaneously

Answer 11

That's right but what it boils down to?

Answer 12

You may like to state what \( A\) and \( B \) is in your question.

Answer 13

@FoolForMath
A= red balls
B= red balls+ black balls
C= blue balls + pink balls
Given that a red ball is drawn (which means A has occurred) , what's the probability that a ball from B is drawn. A is a subset of B
so B has also occurred
so P(B|A)=1

Answer 14

Unless I am going mad, \[ P(B|A) \neq P(A\cap B)/P(B) \]

Answer 15

A is a subset of B
\[P(B/A)= P (A\cap B)/P(A)\]
now
\[P(A\cap B)= P(A)\]
\[P(B/A)= P (A)/P(A)=1\]

Answer 16

Yes, If \( A\subset B \implies P(A\cap B) = P(A)\) that's all we are required in this problem.

Answer 17

Many Thanks for your help. I knew I was missing something that should have been obvious - really appreciate your help.

Answer 18

Glad to help :)