Check whether the following probabilities P(A) and P(B) are consistently defined
(i) P (A) = 0.5, P (B) = 0.7, P (A ∩ B) = 0.6
(ii) P (A) = 0.5, P (B) = 0.4, P (A υ B) = 0.8

Solution:

Two probabilities P(A) and P(B)B are said to be consistently defined if P(A ∩ B) < P(A) and P(A ∩ B) < P(B).

(i) The given probabilities are,

P (A) = 0.5, P (B) = 0.7, P (A ∩ B) = 0.6.

We know that P(A ∩ B) must be less than or equal to P(A) and P(B) for P(A) and P(B) to be said to be consistently defined.

But here, P (A ∩ B) > P (A) though.

Hence, P (A) and P (B) are NOT consistently defined.

(ii) P (A) = 0.5, P (B) = 0.4, P (A υ B) = 0.8

By addition theorem of probability,

P(A ∩ B) = P(A) + P(B) - P(A υ B)

= 0.5 + 0.4 - 0.8

= 0.1

Here, P(A ∩ B) < P(A) and P(A ∩ B) < P(B).

Hence, P (A) and P (B) are consistently defined

NCERT Solutions Class 11 Maths Chapter 16 Exercise 16.3 Question 12

Check whether the following probabilities P(A) and P(B) are consistently defined (i) P (A) = 0.5, P (B) = 0.7, P (A ∩ B) = 0.6 (ii) P (A) = 0.5, P (B) = 0.4, P (A υ B) = 0.8.

Summary:

(i) P (A) and P (B) are NOT consistently defined. (ii) P (A) and P (B) are consistently defined