Q5. Three coins are tossed once. Find the probability of (i) exactly 2 heads, (ii) at least 2 heads, (iii) at most 2 heads, (iv) no tail (NCERT Exercise 14.2, Q8 (ii), (iii), (iv) and (viii)).
Of the 8 equally likely outcomes: (i) HHT, HTH, THH: 3/8. (ii) Those and HHH: 4/8 = 1/2. (iii) All but HHH: 7/8. (iv) Only HHH: 1/8.
Q6. In a lottery, a person chooses uniformly at random six different numbers from 1 to 20, and wins if they match the six numbers fixed in advance (order does not matter). Find the probability of winning (NCERT Exercise 14.2, Q11).
There are 20C6 = 38 760 equally likely choices and one of them wins, so the probability is 1/38 760.
Q7. P(E) = 1/4, P(F) = 1/2 and P(E and F) = 1/8. Find P(E or F) and P(not E and not F) (NCERT Exercise 14.2, Q15).
P(E ∪ F) = 1/4 + 1/2 − 1/8 = 5/8, and P(E′ ∩ F′) = 1 − 5/8 = 3/8.
Q8. Of 60 students, 30 opted for NCC, 32 for NSS and 24 for both. Find the probability that a student chosen at random opted for (i) NCC or NSS, (ii) neither, (iii) NSS but not NCC (NCERT Exercise 14.2, Q21).
(i) (30 + 32 − 24)/60 = 38/60 = 19/30. (ii) 1 − 19/30 = 11/30. (iii) (32 − 24)/60 = 8/60 = 2/15.
Q9. 100 students are split uniformly at random into sections of 40 and 60. Find the probability that you and your friend are in the same section (NCERT Miscellaneous Exercise, Q5).
Number the 100 places, 40 in one section and 60 in the other. You and your friend take two different places in 100 × 99 = 9900 equally likely ordered ways, and both places are in the same section in 40 × 39 + 60 × 59 = 5100 of them. So P(same) = 5100/9900 = 17/33, and P(different) = 16/33.
Q10. Three letters are put at random into three addressed envelopes, one in each. Find the probability that at least one letter is in its proper envelope (NCERT Miscellaneous Exercise, Q6).
Of the 3! = 6 equally likely arrangements, only 2 put every letter in a wrong envelope. So P(at least one right) = 1 − 2/6 = 2/3.
Q11. From 5 persons (Harish M 30, Rohan M 33, Sheetal F 46, Alis F 28, Salim M 41), one is chosen at random. Find the probability that the person is male or over 35 (NCERT Miscellaneous Exercise, Q8).
P(male) = 3/5 and P(over 35) = 2/5, and Salim is both, with 1/5. So P = 3/5 + 2/5 − 1/5 = 4/5.
Q12. A box has 10 red, 20 blue and 30 green marbles, and 5 are drawn uniformly without replacement. Find the probability that (i) all are blue, (ii) at least one is green (NCERT Miscellaneous Exercise, Q1).
(i) 20C5/60C5 = 15 504/5 461 512 = 34/11 977. (ii) No green means all 5 come from the 30 red and blue marbles, so the answer is 1 − 30C5/60C5 = 1 − 117/4484 = 4367/4484.
Q13. In a relay race with five teams A, B, C, D and E, all finishing orders are equally likely. Find the probability that (a) A, B and C finish first, second and third, (b) A, B and C are the first three, in any order (NCERT Miscellaneous Example 12).
The first three places can be filled in 5P3 = 5 × 4 × 3 = 60 equally likely ways. (a) Only ABC: 1/60. (b) The 3! = 6 orders of A, B and C: 6/60 = 1/10.