# Set operations and Computations of Probabilities

Let P(A) denote the probability of a set A, and let A / B and A / B denote the union and intersection, respectively, of two sets A and B. If P(A) = 0.7, P(B) = 0.6, P(C) = 0.3, and P(A / C) = 0.2, find the minimum and maximum possible values of P(A / (B / C)).

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#### Solution Preview

Dear Student,

First, let me state some notation and properties.

Notation:

=> = implies

<= = less than or equal to

A* = the complement of A

A / B = A intersect B

A / B = A union B

A < B = A is contained in B

A / BC = A / (B / C) -- I include parentheses to clarify order of operation.

Properties:

A < B => P(A) <= P(B) -- i.e., ...

#### Solution Summary

The probabilities of three sets A, B, and C are given, along with that of the intersection of A and C. This is used to deduce a range of possible values for a specific composition of all three sets A, B, and C. The basic tools are properties of set composition (intersection, union, and complement), as well as the subadditivity and addition laws of probability.