GMAT Math : Probability inequalities

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:

  1. P(any event) ≤ 1
  2. 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)