Hardy-Weinberg Equilibrium: To compute probability of finding two siblings (random mating)
Are you sure it is 0.59 because I am getting 0.5625 for part (a) and this is how I computed. 

To check genetic similarity of siblings we first have to pick up mating pairs. Six mating pairs possible with following offspring probabilities (choice of first mate x choice of second mate x same offspring genotype freq). 


AA x AA : will always give same type of offspring: prob = 0.25x0.25x1
aa x aa : will always give same type of offspring: prob = 0.25x0.25x1
AA x aa : will always give same type of offspring: prob = 0.25x0.25x1
Aa x Aa : will give Aa in half the number of times: prob = 0.5x0.5x0.5
Aa x AA : will give two types of similar offsprings, half the number of times - AA and Aa: prob = 0.5x0.25x0.5x2
Aa x aa : will give two types of similar offsprings, half the number of times - aa and Aa: prob = 0.5x0.25x0.5x2


Total probability = 3x(0.5x0.5x0.5) + 3x(0.25x0.25) = 0.375+0.1875 = 0.5625

Assumptions:


Population is infinite
Population in equilibrium