Paityn Nielsen

Answered question

2022-03-16

Calculate $P\left[A,B,C\right]$ from $P\left[A,B\right]$ and $P\left[B,C\right]$

Answer & Explanation

Rolloniju9

Beginner2022-03-17Added 2 answers

Step 1
I think you can constrain $P\left[A,B,C\right]$ to an interval such as
$max\left(0,,P\left[A,B\right]+P\left[A,C\right]-P\left[A\right],,P\left[A,B\right]+P\left[B,C\right]-P\left[B\right],,P\left[A,C\right]+P\left[B,C\right]-P\left[C\right]\right)$
$\le P\left[A,B,C\right]\le$
$min\left(P\left[A,B\right],,P\left[A,C\right],,P\left[B,C\right],,1+P\left[A,B\right]+P\left[A,C\right]+P\left[B,C\right]-P\left[A\right]-P\left[B\right]-P\left[C\right]\right)$
As an example, if $P\left[A\right]=P\left[B\right]=P\left[C\right]=\frac{1}{2}$ and $P\left[A,B\right]=P\left[A,C\right]=P\left[B,C\right]=\frac{1}{4}$ then $0\le P\left[A,B,C\right]\le \frac{1}{4}$

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