Finding the least upper bound of the probability of two events

Question

If $P[A|B]=1/3$ and $P[B|A]=1/3$, what is the largest possible value for $P[A\,and\,B]$? (Find the least upper bound).

I know that $P[A|B]*P[B] = P[A\,and\,B]$, and likewise $P[B|A]*P[A] = P[A\,and\,B]$. But how do I get a numerical value for the value of $P[A\,and\,B]$?

Answer 1

Set $P(A|B)P(B) = P(B|A)P(A)$ from your two equations to get $P(A) = P(B)$.

Now $P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and } B)$, or $P(A\text{ or }B) = 2P(A)-\frac{1}{3}P(A)$. Hence $\frac{5}{3}P(A) \leq 1$, so $P(A) \leq \frac{3}{5}$.

Hence $P(A\text{ and }B) = \frac{1}{3}P(A) \leq \frac{1}{5}$. It is not too difficult to construct events where this bound is achieved.