If A and B are independent events and $P(A) = \frac{1}{2}$ $P(B) = \frac{1}{3}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(A \cap B)$ | (I) $\frac{1}{2}$ | | (B) $P(\bar{A})P(B) + P(A)P(\bar{B})$ | (II) $\frac{1}{3}$ | | (C) $P(A \mid B) + P(B \mid A)$ | (III) $\frac{1}{6}$ | | (D) $P(A \cap \bar{B})$ | (IV) $\frac{5}{6}$ | Choose the correct answer from the options given below:

If a fair coin is tossed 10 times, then the probability of getting all heads or all tails, is :

The binomial distribution for which the mean is 5 and variance 4, is

In Binomial distribution with parameters $n = 12$ and $p = \frac{1}{3}$, value of $E(X^2) + E(X)$ is :

In Binomial distribution with parameters $n = 100$ and p, Variance of distribution is maximum when p is equal to :

A doctor is to visit a patient. It is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he arrives late.

The probability that he comes by bus is:

Let box I contains 3 black and 4 white balls, box II contains 2 black and 2 white balls, box III contains 4 black and 3 white balls. A box is selected at random and then a ball is randomly drawn from the selected box. If the color of the ball is black then the probability that the ball is drawn from box III, is: