# What is P ( A &#x222A;<!-- ∪ --> ( B &#x2229;<!-- ∩ --> C ) ) ?

What is $P\left(A\cup \left(B\cap C\right)\right)$?

The question says it all. I know
$P\left(A\cap \left(B\cup C\right)\right)=P\left(A\cap B\right)+P\left(A\cap C\right).$
Would this mean,
$P\left(A\cup \left(B\cap C\right)\right)=P\left(A\cup B\right)+P\left(A\cup C\right)?$
Just want to make sure.
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Gillian Kelly
The equation
$P\left(A\cup \left(B\cap C\right)\right)=P\left(A\cup B\right)+P\left(A\cup C\right)$
will almost certainly not hold. This is because
$P\left(A\cup \left(B\cap C\right)\right)=P\left(\left(A\cup B\right)\cap \left(A\cup C\right)\right)\le P\left(A\cup B\right)$
and similarly
$P\left(A\cup \left(B\cap C\right)\right)\le P\left(A\cup C\right).$
For the stated equation to hold, both $P\left(A\cup B\right)$ and $P\left(A\cup C\right)$ would have to be zero, showing that
$P\left(A\right)=P\left(B\right)=P\left(C\right)=0.$
However, if this condition is satisfied, then the stated equation will hold.