Law of Total Probability

Let $S=\sum _{k=1}^n{B}_k$ , where $B_i\cap B_j=\varnothing$ if $i\ne j$, then for any event $A$, $P(A)=\sum _{R=1}^{n}P(A|B_R)\cdot P(B_R)$.

Example

You have one fair coin and a biased coin that lands on heads with a probability of $\frac{3}{4}$ . If you pick a random coin and toss it three times, find the probability that all three tosses are heads.

Draw the probability tree.

$P(A)=P(A|B_1)\cdot P(B_1)+P(A|B_2)\cdot P(B_2)$ $P(A)=(\frac{1}{2}{)}^3\cdot 0.5+(\frac{3}{4}{)}^3\cdot 0.5$

LOTP with Conditioning

This is an extension to LOTP introduced in CS287.

$P(x)={\sum }_{y}P(x,y)$ $P(x|z)={\sum }_{y}P(y|z)P(x|y,z)$