Probability (Class 12)

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 .

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 .

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 .