(a) A die with number 1 to 6 is biased such that P (2)=3/10 and probability of other numbers is equal. Find the mean of the number of times number 2…

A die with number $\displaystyle 1$ to $\displaystyle 6$ is biased such that $\displaystyle \mathrm{P}(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number $\displaystyle 2$ appears on the dice, if the dice is thrown twice.

Two dice are thrown. Defined are the following two events A and B: $\displaystyle \mathrm{A}=\{(x, \mathrm{y}): x+\mathrm{y}=9\}, \mathrm{B}=\{(x, \mathrm{y}): x \neq 3\}$, where $\displaystyle (x, \mathrm{y})$ denote a point in the sample space. Check if events A and B are independent or mutually exclusive.