Probability (Class 12)

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let E1 be the event that the ball chosen belonged to Box I and let E2 be the event that the ball chosen belonged to Box II. Let F1 be the event that the ball chosen is red and let F2 be the event that the ball chosen is green. Then which of the following statements is (are) TRUE ?

1. The events E1 and F1 are independent
2. The events E2 and F2 are dependent
3. The conditional probability P(F1 E1) is equal to the conditional probability P(F1 E2)
4. The conditional probability P(F1 E1) is greater than the conditional probability P(F2 E2)

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let X be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If is the mean of the random variable X, then the value of 77 is .

Three students S1, S2, and S3 are given a problem to solve. Consider the following events: U: At least one of S1, S2, and S3 can solve the problem, V: S1 can solve the problem, given that neither S2 nor S3 can solve the problem, W: S2 can solve the problem and S3 cannot solve the problem, T: S3 can solve the problem. For any event E, let P(E) denote the probability of E. If P(U) = 12, P(V) = 110, and P(W) = 112, then P(T) is equal to

1. 1336
2. 13
3. 1960
4. 14

A factory has a total of three manufacturing units, M1, M2, and M3, which produce bulbs independent of each other. The units M1, M2, and M3 produce bulbs in the proportions of 2:2:1, respectively. It is known that 20\% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by M1, 15\% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by M2 is 25. If a bulb is chosen randomly from the bulbs produced by M3, then the probability that it is defective is .

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is 12. Also assume that the probability of the answer for a question being guessed, given that the student’s answer is correct, is 16. Then the probability that the student knows the answer of a randomly chosen question is

1. 112
2. 17
3. 57
4. 512

Let X be a random variable, and let P(X=x) denote the probability that X takes the value x. Suppose that the points (x,P(X=x) ), x=0,1,2,3,4, lie on a fixed straight line in the xy-plane, and P(X=x)=0 for all x R-\0,1,2,3,4\. If the mean of X is 52, and the variance of X is , then the value of 24 is .

A bag contains N balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For i=1,2,3, let Wi, Gi, and Bi denote the events that the ball drawn in the ith draw is a white ball, green ball, and blue ball, respectively. If the probability P(W1 G2 B3)= 25N and the conditional probability P(B3 W1 G2)= 29, then N equals .