I play tennis with my parents; the chances for me to win a game against $\operatorname{Mum}(M)$ are $p$ and against Dad $(D) q$, where $0<q<p<1$. We agreed to have three games, and their order can be $D M D$ (where I play against Dad, then Mum then again Dad) or $M D M$. The results of games are independent.

Calculate under each of the two orders the probabilities of the following events:

a) that I win at least one game,

b) that I win at least two games,

c) that I win at least two games in succession (i.e., games 1 and 2 or 2 and 3 , or 1 , 2 and 3$)$,

d) that I win exactly two games in succession (i.e., games 1 and 2 or 2 and 3 , but not 1,2 and 3 ),

e) that I win exactly two games (i.e., 1 and 2 or 2 and 3 or 1 and 3 , but not 1,2 and 3$)$.

In each case a)- e) determine which order of games maximizes the probability of the event. In case e) assume in addition that $p+q>3 p q$.