Q5. P(A) = 1/4, P(B) = 1/2 and P(A ∩ B) = 1/8. Find P(not A and not B) (NCERT Exercise 13.2, Q9, page 422).
NCERT set these numbers in Class 11 too (Exercise 14.2, Q15). What is new: P(A ∩ B) = 1/8 = (1/4)(1/2), so A and B are independent, and P(A′ ∩ B′) = (3/4)(1/2) = 3/8. The addition rule agrees: 1 − (1/4 + 1/2 − 1/8) = 3/8.
Q6. Two balls are drawn with replacement from a box with 10 black and 8 red balls. Find the probability that (i) both are red; (ii) the first is black and the second red; (iii) one is black and the other red (NCERT Exercise 13.2, Q13, page 422).
With replacement the draws are independent. (i) (8/18)2 = 16/81. (ii) (10/18)(8/18) = 20/81. (iii) Either order: 2 × 20/81 = 40/81.
Q7. In a hostel, 60% read a Hindi newspaper, 40% an English one and 20% both. For a student chosen at random, find (a) P(neither); (b) P(reads English | reads Hindi); (c) P(reads Hindi | reads English) (NCERT Exercise 13.2, Q16, page 423).
(a) 1 − (0.6 + 0.4 − 0.2) = 0.2 = 1/5. (b) 0.2/0.6 = 1/3. (c) 0.2/0.4 = 1/2. Since 0.2 ≠ 0.6 × 0.4, the two habits are not independent.
Q8. The probability of an even prime number on each die, when a pair of dice is rolled, is (A) 0; (B) 1/3; (C) 1/12; (D) 1/36 (NCERT Exercise 13.2, Q17, page 423).
The only even prime is 2, so (1/6)(1/6) = 1/36: (D).
Q9. Two events A and B are independent if (A) A and B are mutually exclusive; (B) P(A′B′) = [1 − P(A)][1 − P(B)]; (C) P(A) = P(B); (D) P(A) + P(B) = 1 (NCERT Exercise 13.2, Q18, page 423).
(B): it says A′ and B′ are independent, which holds exactly when A and B are.
Q10. Show that if E and F are independent, then E′ and F′ are independent.
P(E′ ∩ F′) = 1 − P(E ∪ F) = 1 − P(E) − P(F) + P(E) P(F) = (1 − P(E))(1 − P(F)) = P(E′) P(F′).
Q11. Beyond the NCERT text (JEE): two coins are tossed. A is “first coin head”, B is “second coin head” and C is “both coins show the same face”. Show that each pair is independent but the three are not.
Each event has probability 1/2, and each pair meets only in HH: probability 1/4 = (1/2)(1/2). But A ∩ B ∩ C = {HH} also has probability 1/4, not (1/2)3 = 1/8.
Q12. Each entry of a 2 × 2 determinant is 0 or 1, chosen independently, each with probability 1/2. Find the probability that the determinant is positive (NCERT Miscellaneous Exercise on Chapter 13, Q8, page 436).
ad − bc > 0 needs ad = 1 and bc = 0. The entries are independent, so the probability is (1/4)(3/4) = 3/16.