Probabilities of solving a specific mathematical problem independently by three students are $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$. If all the three students try to solve the problem independently, the probability that the problem is solved, is:

If E and F are two independent events such that $P(E)=\dfrac{1}{3}, P(F)=\dfrac{3}{5}$, then which of the following is/are correct ?

A. $P(E \cap F) = \dfrac{1}{5}$

B. $P\left(\dfrac{\overline{E}}{F}\right) = \dfrac{2}{3}$

C. $P\left(\dfrac{E}{\overline{F}}\right) = \dfrac{2}{3}$

D. $P\left(\dfrac{E}{F}\right) = \dfrac{2}{3}$

Choose the correct answer from the options given below:

Let X be a random variable. Let E (X) and Var (X) denote the mean and the variance of X respectively. Then match List-I with List-II

Choose the correct answer from the options given below:

Five bad apples are mixed with 15 good apples accidently. Let the random variable X be the number of bad apples in a draw of two apples, then mean of X is

If $A$ and $B$ are independent events such that $0 < P(A) < 1$ and $0 < P(B) < 1$, then identify the correct statements.

(A) $A$ and $B'$ are independent

(B) $A'$ and $B$ are independent

(C) $A$ and $B$ are mutually exclusive

(D) $A'$ and $B'$ are independent

Choose the correct answer from the options given below :

If $P(A) = \frac{3}{10}$, $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{3}{5}$ then the value of $P(B|A) + P(A|B)$ is:

Three bags contain a number of red and white balls as follows:

Bag I: 3 red balls

Bag II: 2 red balls and 1 white ball

Bag III: 3 white balls

The probability that bag $i$ will be chosen, and a ball is selected from it is $\frac{i}{6}, i=1,2,3$. The probability that a red ball is selected is equal to: