Probability based inequalities is a topic often tested by the GMAT Quant section as you head towards a Q50+

Let’s start with an example to get warmed up :

**The probability of it raining on a particular day is 0.6 and the probability of it being windy is 0.7. Let x denote the probability that it will both rain and be windy on the same day. What is always true for x?**

**A. 0.1 ≤ x ≤ 0.3**

**B. 0.3 ≤ x ≤ 0.6**

**C. 0.6 ≤ x ≤ 0.7**

**D. 0.4 ≤ x ≤ 0.5**

**E. 0.3 ≤ x ≤ 0.5**

**Here is a refresher on probability inequalities:**

**P(any event) ≤ 1****P(A AND B) ≤ P(A) and P(A AND B) ≤ P(B)**

The probability of two or more events happening together at the same time is less than the probability of each happening unrestricted. Think about it : We can claim more confidently that ** a classroom contains a male student** as opposed to claiming it contains

**a male student interested in playing the violin who received 99 in science in the last exam.**The above two rules form the core probability inequalities. We derive another critical inequality on the basis of the above two which is frequently tested in hard quant questions :

We know :- P(A U B) = P(A) + P(B) – P(A AND B) : this is the set union rule in probability

Now, P(A U B) ≤ 1 (as with any probability, P(A U B) is upper bounded by 1)

Hence, P(A) + P(B) – P (A AND B) ≤ 1

P(A AND B) ≥ P (A) + P (B) – 1

Also, P(A AND B) ≤ P(A), P(B) from rule 2 above.

Hence, **P(A) + P(B) – 1 ≤ P(A AND B) ≤ P(A), P(B)**

**This is a critical inequality and we urge readers to get familiar with it** 🙂

Coming to the question,

P(A) = 0.6 and P(B) = 0.7

So, 0.6 + 0.7 – 1 ≤ P(A AND B) ≤ 0.6, 0.7

0.3 ≤ P(A AND B) ≤ 0.6

0.3 ≤ x ≤ 0.6

We go with **Option (C)**