Three fair dice are thrown. What is the probability of getting a total of 15 given that they exhibit three different numbers that are in arithmetic progression?

A person writes four letters and address four envelopes. If the letters are placed in the envelopes at random, then the probability that not all letters are placed in the right envelope is

Let $A$ and $B$ be two events such that one of the two events must occur. Given that the chance of occurrence of $A$ is $\frac{2}{3}$ the chance of occurrence of $B$, then odds in favour of $B$ is

Vasant and Jothi play a game with a coin. Vasant stakes ₹ 1 and throw the coins four times. If he throws four heads, he gets his stake and ₹3 from Jothi. If he throws only three heads and they are consecutive, he gets his stake and ₹2 from Jothi. If he throws only two heads and they are consecutive, he gets his stake and ₹1 from Jothi. In all other cases Jothi takes the stake money. Find the expectation of Vasant's gain.