A lower bound on an intersection

Topic: Random experiments and probability · more questions on this topic
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Let \mathbb{P}(A)=0.9 and \mathbb{P}(B)=0.8. Show that \mathbb{P}(A\cap B)\ge 0.7.

Worked answer (click to show)

The union formula rearranges to \mathbb{P}(A\cap B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cup B). The union has probability at most 1, so

\mathbb{P}(A\cap B)\ge 0.9+0.8-1=0.7.