Comprehension Based Questions: Probability - Free MCQ Practice Test

PASSAGE - 1

There are n urns, each of these contain n + 1 balls. The ith urn contains i white balls and (n + 1 – i) red balls. Let u_i be the event of selecting ith urn, i = 1, 2, 3 .........., n and w the event of getting a white ball.

Q. If P(u_i) ∝ i, where i = 1, 2, 3,......., n, then P(w) =(2006 - 5M, –2)

PASSAGE - 1

There are n urns, each of these contain n + 1 balls. The ith urn contains i white balls and (n + 1 – i) red balls. Let u_i be the event of selecting ith urn, i = 1, 2, 3 .........., n and w the event of getting a white ball.

Q. If P(u_i) = c, (a constant) then P(u_n/w) = (2006 - 5M, –2)

PASSAGE - 1

There are n urns, each of these contain n + 1 balls. The ith urn contains i white balls and (n + 1 – i) red balls. Let u_i be the event of selecting ith urn, i = 1, 2, 3 .........., n and w the event of getting a white ball.

Q. Let P(u_i) = , if n is even and E denotes the event of choosing even numbered urn, then the value of P(w / E) is (2006 - 5M, –2)

PASSAGE - 2
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. (2009)

Q. The probability that X = 3 equals

PASSAGE - 2
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. (2009)

Q. 5. The probability that X ≥ 3 equals

PASSAGE - 2
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. (2009)

Q.The conditional probability that X ≥ 6 given X > 3 equals

PASSAGE - 3
Let U₁ and U₂ be two urns such that U₁ contains 3 white and 2 red balls, and U₂ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U₁ and put into U₂.
However, if tail appears then 2 balls are drawn at random from U₁ and put into U₂ . Now 1 ball is drawn at random from U₂. (2011)

Q. The probability of the drawn ball from U₂ being white is

PASSAGE - 3
Let U₁ and U₂ be two urns such that U₁ contains 3 white and 2 red balls, and U₂ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U₁ and put into U₂.
However, if tail appears then 2 balls are drawn at random from U₁ and put into U₂ . Now 1 ball is drawn at random from U₂. (2011)

Q. Given that the drawn ball from U₂ is white, the probability that head appeared on the coin is

PASSAGE - 4
A box B₁ contains 1 white ball, 3 red balls and 2 black balls. Another box B₂ contains 2 white balls, 3 red balls and 4 black balls. A third box B₃ contains 3 white balls, 4 red balls and 5 black balls.

Q. If 1 ball is drawn from each of the boxes B₁, B₂ and B₃, the probability that all 3 drawn balls are of the same colour is (JEE Adv. 2013)

PASSAGE - 4
A box B₁ contains 1 white ball, 3 red balls and 2 black balls. Another box B₂ contains 2 white balls, 3 red balls and 4 black balls. A third box B₃ contains 3 white balls, 4 red balls and 5 black balls.

Q. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box B₂ is

PASSAGE - 5
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let x_i be number on the card drawn from the ith box, i = 1, 2, 3. (JEE Adv. 2014)

Q. The probability that x₁ + x₂ + x₃ is odd, is

PASSAGE - 5
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let x_i be number on the card drawn from the ith box, i = 1, 2, 3. (JEE Adv. 2014)

Q. 12. The probability that x₁, x₂, x₃ are in an arithmetic progression, is

PASSAGE - 6
Let n₁ and n₂ be the number of red and black balls, respectively, in box I. Let n₃ and n₄ be the number of red and black balls, respectively, in box II. (JEE Adv. 2015)

Q. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is , then the correct option(s) with the possible values of n1, n₂, n₃ and n₄ is(are)  

PASSAGE - 6
Let n₁ and n₂ be the number of red and black balls, respectively, in box I. Let n₃ and n₄ be the number of red and black balls, respectively, in box II. (JEE Adv. 2015)

Q. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is , then the correct option(s) with the possible values of n₁ and n₂ is(are)

PASSAGE - 7
Football teams T₁ and T₂ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T₁ winning, drawing and losing a game against T₂ are    and   respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game.
Let X and Y denote the total points scored by teams T₁ and T₂ respectively after two games.

Q.P (X > Y) is (JEE Adv. 2016)

PASSAGE - 7
Football teams T₁ and T₂ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T₁ winning, drawing and losing a game against T₂ are    and   respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game.
Let X and Y denote the total points scored by teams T₁ and T₂ respectively after two games.

Q. P (X = Y) is (JEE Adv. 2016)