$.[20$ Points$][$MID$]$ Consider a genetic algorithm using a vector of integer representation with vectors

of length $5.$ The fitness is to be maximized. Assume that the initial population was as follows:

Individual Genotype Fitness

$15,2,5,3,110$

$24,2,3,3,320$

$33,4,1,2,510$

$41,4,3,2,530$

$($a$)$ What is each individual$$s probability of being selected as parent if proportional $($roulette wheel$)$

selection is used?

$($b$)$ What is each individual$$s probability of being selected as parent if deterministic binary

tournament selection is used $($in which two distinct individuals compete in any tournament and

ties are broken randomly$)?$

$($c$)$ If we select individuals $3$ and $4$ as parents for N point crossover for some value of N$,$ can the

genotype of a resulting child be $3,4,3,2,5?$ how about $1,4,1,2,3?$ Briefly justify your answer