If this question has been answered somewhere else, I apologize, and welcome the redirection.
I've been given the formula for conditional probability (the probability of event A happening, given event B) as: $$P(A | B) = \frac{P(A\text{ and }B)}{P(B)}$$
My question is if $P(A\text{ and }B) = P(A) * P(B)$ then wouldn't the probability of event B just cancel each other out in the numerator and denominator?
In my head, I know this can't be right, so I think I must have made a mistake when finding the probability of (A and B) but I'm not sure where I went wrong…or what formula I should be using instead.
Thanks in advance for any help/guidance!
Best Answer
$P(A \text{ and }B)=P(A)P(B)$ if and only if the events $A$ and $B$ are independent. For instance, consider flipping a coin, and let $A$ be the event that it lands on heads, and let $B$ be the event that it lands on tails. Clearly $P(A \text{ and }B)=0$, but $P(A)P(B)=1/4$.