Three vertices are chosen randomly from the nine vertices of a regular 9-sided polygon. The probability that they form the vertices of an isosceles triangle, is

If $$A, B$$ and $$C$$ are mutually exclusive and exhaustive events of a random experiment such that $$P(B)=\frac{3}{2} P(A)$$ and $$P(C)=\frac{1}{2} P(B)$$, then $$P(A \cup C)$$ equals to

In a trial, the probability of success is twice the probability of failure. In six trials, the probability of at most two failure will be

A die is thrown twice and the sum of numbers appearing is observed to be 8 . What is the conditional probability that the number 5 has appeared atleast once?