If A and B are independent events such that $P(A|B) = \frac{1}{3}$ and $P(B) = \frac{1}{2}$, then the value of $P(A \cap B)$ is equal to

The value of $k$ 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:

A die is thrown three times. If the first throw is a five, the probability of getting 14 as the sum is

If X is a random variable and a, b are real numbers, then which of the following statements are correct?

(A) $E[aX+b] = a E(X) + b$

(B) $Var (aX + b) = a^2 Var (X) + b$

(C) $Var (aX + b) = a Var (X)$

(D) $Var (X) = E(X^2) - [E(X)]^2$

Choose the correct answer from the options given below:

A die is thrown and a coin is tossed. The probability that the outcome will be a number greater than 4 or a head or both is

The random variable X has the following probability distribution

such that E(x²) = 2E(x), then the value of b is: