When is $\mathbb P(A\cap B)$ minimale and maximale ? I said that $\mathbb P(A\cap B)$ is maximal when $A=B$ and minimal when $A$ and $B$ are independent. But the thing is I'm not sure that $\mathbb P(B\mid A)\geq \mathbb P(B)$ is true or not, that's wha I have doubt. Indeed, $\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B\mid A)$. But do we have $\mathbb P(B\mid A)\geq \mathbb P(B)$ or not ? And if yes, why ?
When is $\mathbb P(A\cap B)$ minimale and maximale?
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probability
1 Answers
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The probability $\mathbb P(A\cap B)$ is minimal (actually, zero) if $A$ and $B$ are complementary events, like $A$ having heads and $B$ having tails when throwing a coin once. Note that these events are not independent: $\mathbb P(A\cap B)=\mathbb P(\emptyset)=0$, but $\mathbb P(A)\cdot \mathbb P( B)=\frac{1}{4}$. This also gives a counterexample to your second question: We have $\mathbb P(B)=1/2$, but $\mathbb P(B|A)=0$.
Edit: And you are correct: $\mathbb P(A\cap B)$ is maximal when the events $A$ and $B$ coincide.