Question 1206319
i think this works like this.


law of total probability states that p(A) = p(A intersect B) plus p(A intersect complement of B.


you are given:


p(S1) = .7
p(S2) = .3


you are given that p(A given S1) = .2.
you are given that p(A given S2) = .1


p(A given S1) is equal to p(A intersect S1) / p(S1).
this becomes .2 = p(A intersect S1) / .7
solve for p(A intersect S1) to get:
p(A intersect S1) = .7 * .2 = .14


p(A given S2) is equal to p(A intersect S1) / p(S2).
this becomes .1 = p(A intersect S1) / .3
solve for p(A intersect S2) to get:
p(A intersect S2) = .1 * .3 = .03


p(A) is equal to p(A intersect S1) + p(A intersect S2) = .14 + .03 = .17


that should be your answer.


here's a reference.
<a href = "https://www.statisticshowto.com/total-probability-rule/" target = "_blank">https://www.statisticshowto.com/total-probability-rule/</a>


i drew a tree diagram (not very pretty) that shows the same result.
see below.


<img src = "http://theo.x10hosting.com/2024/022901.jpg">


in the diagram, that's p(S1 and S2) and p(A and not A).
the top of the tree is marked as 1, meaning it's 100% probability on top of the tree.
that encompasses all that's beneath it.