Question $4.[15$ points$]$

Scenario Description:

In a computational experiment designed by Prof. Alghoul, there are two circular queues, Q$1$and Q$2,$ each capable of storing integer numbers. Joey, a participant in this game, is given a set of $100$ integer numbers by Prof. Alghoul $-$ specifically, $50$ numbers are even and $50$ are odd. Joey is required to distribute these $100$ integers between Q$1$ and Q$2.$

Once Joey has allocated the integers, Prof. Alghoul initiates an experiment where he selects either Q$1$ or Q$2$ at random and applies a rotate$()$ operation on the chosen queue a random number of times. The rotate$()$ operation moves the front element of the queue to the rear.

If the integer at the front of the queue after the final rotation is odd, Prof. Alghoul wins; if it's even, Joey wins.

Questions:

a$)$ How should Joey distribute the numbers between Q$1$ and Q$2$ to maximize the chances of winning? $(10$ points$)$

b$)$ Given this allocation strategy, what is the probability of Joey winning the game? $(5$points$)$