I am currently reviewing conditional probability in my textbook and have stumbled across this:
The formula for conditional probability is this: Where $A$ and $B$ are two events where $P(B) > 0$ (naturally)
$$P(A \mid B) = \frac{P(A\cap B)}{P(B)}$$
so naturally: $P(A\cap B) = P(B)\cdot P(A\mid B)$
and if $A$ and $B$ are independent, $P(A \mid B) = P(A)$
so $P(A\cap B) = P(B) \cdot P(A)$ which matches the definition of independence.
What I wonder is this. Say that $P(B)$ and $P(A \mid B)$ are non-zero probabilities and $P(A \mid B) \neq P(A)$ so $P(A\cap B) \neq 0$ and $A$ and $B$ are non-disjoint AND not independent events. Is this legal? Have I broken a law of probability? The textbook doesn't specify. If it's legal can someone give me an example in real life of two events that are non-disjoint and not independent?
Also can someone answer my second question?
Does the RHS of $P(A\cap B) = P(B)\cdot P(A\mid B)$ mean that event $B$ occurs first and THEN event $A$ occurs in which event $B$ occurring first affected the probability of event $A$ also occurring?
OR
Does the RHS mean that event $B$ and $A$ occur simultaneously? (If it's this second interpretation, can someone explain how events $A$ and $B$ occurring simultaneously affects event $A$'s probability of occurring?)
Best Answer
Any example where $A \subsetneq B$ (with $P(A) > 0$ and $P(B) < 1$) will satisfy your conditions:
$P(A \mid B)$ should be interpreted as the probability of $A$ occurring if you already know that $B$ occurred. So you can interpret $P(B) \cdot P(A \mid B)$ as first accounting for the probability that $B$ occurred, and then accounting for the probability that $A$ occurred given that you already know $B$ occurred. This is common in sequential computations, i.e. the probability that two draws without replacement from a deck are both aces is $P(\text{first card is ace}) P(\text{second card is ace} \mid \text{first card is ace}) = \frac{4}{52} \cdot \frac{3}{51}$.