Thread: P(A n B) \leq P(A)

1. P(A n B) \leq P(A)

Prove:

$P(A\cap B)\leq P(A)$

using the axioms of probability.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)\geq 0$

$P(A\cap B)\leq P(A)+P(B)$

Now, I am stuck with a P(B).

2. Hint :

$A\cap B\subset A$

Fernando Revilla

3. Suppose that $C\subseteq D$.
Then $\mathcal{P}(D)=\mathcal{P}(C)+\mathcal{P}(D\cap C^c)\ge \mathcal{P}(C)$.
Probability is monotone.