JEE Advanced Physics: Probability questions with solutions

Q1. Two urns contain coloured balls: Urn-I has 4 red and 2 white balls; Urn-II has 2 red and 4 white balls. One ball is drawn at random from Urn-I. If it is red, it is placed in Urn-II; if it is white, it is returned to Urn-I along with one extra white ball taken from Urn-II. Next, one ball is drawn at random from Urn-II. If it is red, it is returned to Urn-II along with one extra red ball taken from Urn-I; if it is white, it is placed in Urn-I. After this pair of operations, it is observed that either the number of white balls in Urn-I is the same as initially, OR the number of red balls in Urn-I is the same as initially. Let p be the probability that the number of red balls in Urn-I remained the same as initially. Find the value of 17p.

1. 5
2. 7
3. 9
4. 12

Answer: 7

There are four equiprobable-structured paths. In Path A (Red then Red from Urn-II) Urn-I ends with 2R 2W — white count preserved. In Path D (White then White from Urn-II) Urn-I ends with 4R 4W — red count preserved. These are the only two paths satisfying the given condition, so the conditional probability p of red being unchanged equals P(Path D) / (P(Path A) + P(Path D)).

Q2. A ship has three independent engines E1, E2, and E3 that operate with probabilities 1/2, 1/4, and 1/4 respectively. The ship is operational if at least two engines function. Let X be the event that the ship is operational. Which of the following are true?

1. P(E1 does not function | ship is operational) = 3/16
2. P(exactly two engines are functioning) = 7/8
3. P(ship is operational | E2 is functioning) = 5/16
4. P(ship is operational | E1 is functioning) = 7/16

Answer: P(ship is operational | E1 is functioning) = 7/16

P(X) = (1/2)(1/4)(1/4) + (1/2)(1/4)(3/4) + (1/2)(3/4)(1/4) + (1/2)(1/4)(1/4) = 1/32 + 3/32 + 3/32 + 1/32 = 8/32 = 1/4. Then evaluate each option using conditional probability.