Q5. Two groups compete for a board. The first wins with probability 0.6 and the second with 0.4. A new product is introduced with probability 0.7 if the first group wins and 0.3 if the second wins. A new product was introduced. Find the probability that the second group introduced it (NCERT Exercise 13.3, Q9, page 432).
The product paths are 0.6 × 0.7 = 0.42 and 0.4 × 0.3 = 0.12, so the probability is 0.12/0.54 = 2/9.
Q6. A girl throws a die. On 5 or 6 she tosses a coin three times; on 1, 2, 3 or 4 she tosses it once. Assume a fair die and independent fair coin tosses. She gets exactly one head. Find the probability that she threw 1, 2, 3 or 4 (NCERT Exercise 13.3, Q10, page 432).
P(one head | 1–4) = 1/2 and P(one head | 5 or 6) = 3/8. The paths are (2/3)(1/2) = 1/3 and (1/3)(3/8) = 1/8, so the probability is (1/3)/(1/3 + 1/8) = 8/11.
Q7. Operators A, B and C make 1%, 5% and 7% defective items and are on the job 50%, 30% and 20% of the time. Assume equal output rates, so the time shares also give the fractions of items produced. A defective item is selected at random. Find the probability that A produced it (NCERT Exercise 13.3, Q11, page 432).
The defective paths are 0.005, 0.015 and 0.014, which add up to 0.034. So the probability is 0.005/0.034 = 5/34.
Q8. A uniformly random card is lost from a pack of 52. Two cards drawn from the rest are both diamonds. Find the probability that the lost card is a diamond (NCERT Exercise 13.3, Q12, page 432).
If the lost card is a diamond (probability 13/52), two diamonds come from 12 of 51 cards; if not (39/52), from 13 of 51. The chances are in the ratio (1/4)(12 × 11) : (3/4)(13 × 12) = 33 : 117, so the probability is 33/150 = 11/50.
Q9. Three boxes hold two gold coins, two silver coins, and one of each. A box is chosen at random and a coin taken out is gold. Find the probability that the other coin in the box is gold (NCERT Example 17, pages 426–427).
The other coin is gold only for the gold–gold box. The gold paths are (1/3)(1) and (1/3)(1/2), so the probability is (1/3)/(1/2) = 2/3, not 1/2: the gold–gold box is twice as likely to give a gold coin.
Q10. Beyond the NCERT text (JEE): in NCERT Example 18, the person is tested a second time and is again positive. Assume the two results are independent for a person with HIV and for a person without it. Find the new probability of HIV.
The two-positive paths are 0.001 × 0.92 = 0.00081 and 0.999 × 0.012 = 0.0000999, so the probability is 0.00081/0.0009099 = 300/337, about 0.89. The first answer, about 0.083, has become the prior for the second test.
Q11. 5% of men and 0.25% of women have grey hair, and there are equal numbers of men and women. A grey-haired person is chosen at random. Find the probability that the person is male (NCERT Miscellaneous Exercise on Chapter 13, Q3, page 435).
(1/2)(0.05)/
Q12. In this hypothetical probability model, the chance of a heart attack is 40%. A yoga course cuts the risk by 30% relative to the original risk and a drug by 25% relative to it, and a patient chooses one of the two with equal probability. A patient has a heart attack. Find the probability that the patient did the yoga course (NCERT Miscellaneous Exercise on Chapter 13, Q7, page 436).
The risks are 0.4 × 0.7 = 0.28 and 0.4 × 0.75 = 0.30, each with prior 1/2, so the probability is 0.28/0.58 = 14/29.
Q13. Bag I has 3 red and 4 black balls, and Bag II has 4 red and 5 black. One ball is moved from Bag I to Bag II, and then a ball drawn from Bag II is red. Find the probability that the moved ball was black (NCERT Miscellaneous Exercise on Chapter 13, Q10, page 436).
If the moved ball was red (3/7), Bag II has 5 red of 10; if black (4/7), 4 red of 10. So the probability is (4/7)(4/10)/