I am new in probability. Sorry if I am asking stupid questions. I have a simple question:
Proof that $$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$
for P(A) and P(B) not equal to zero.
I have googled the answer which should be: $$P(A|B)=\frac{P(A ∩ B)}{P(B)}$$ $$P(B|A)=\frac{P(A ∩ B)}{P(A)}$$ $$P(A∣B)P(B)=P(A∩B)=P(B∣A)P(A) $$ Dividing both sides by P(B): $$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$
My question is that I have done something different and I what to know whether my answer is acceptable or not.
My answer: $$P(B|A)=\frac{P(A ∩ B)}{P(A)}$$ $$P(B|A)P(A)=P(A ∩ B)$$
and $$P(A|B)=\frac{P(A ∩ B)}{P(B)}$$ Substitute $P(A ∩ B)=P(B|A)P(A)$ in we have: $$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$
This was actually one of my exam questions with 4 marks and I am just thinking how many marks am I going to get with my answer. Thanks all.
At the end of the day, only your teacher can tell you how points will be distributed. However, your approach is correct, and I can't see any reason you should be docked for it.