If A is any event associated with sample space and If $E_1, E_2, E_3$ are mutually exclusive and exhaustive events. Then which of the following are true? (A) $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$ (B) $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$ (C) $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{i=1}^3 P(A|E_i)P(E_i)}$, $i = 1,2,3$ (D) $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$, $i = 1,2,3$ Choose the correct answer from the options given below:

Let A, B, C be three events. If the probability of occurring exactly one out of A and B is $\frac{3}{5}$, exactly one of B and C is $\frac{1}{5}$, exactly one of C and A is $\frac{3}{5}$ and that of occurring of three events is $\frac{4}{25}$, then the probability of occurring at least one of them is

Let E and F are events associated with an experiment. If $P(E) = 0.4$, $P(F) = 0.8$ and $P(F|E) = 0.6$, then $P(E|F)$ is

If A and B are independent events, then which of the following statements are TRUE?

(A) $P(A \cap B) = P(A).P(B)$

(B) $P(A \cap B) = P(A) - P(B)$

(C) $P(A \cup B) = P(A) + P(B) - P(A).P(B)$

(D) $P(A \cap B) = P(A). P(B|A)$

Choose the correct answer from the options given below:

In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs. 1 otherwise, when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. The expected value of the amount(in Rs.) he wins or loses is:

A die is tossed 6 times and getting "1 or 5" is considered a success. The probability of getting at least one success in six tosses is:

The probability distribution of a discrete random variable X is given as :

The value of K is: