The gene Prdm9 is thought to regulate hotspots of recombination (crossing over) in mammals, including humans. In the people of Han Chinese descent living in the Los Angeles area there are five alleles at the Prdm9 gene, labeled A₁,A₂,A₃.A₄, and A₅ .

The relative frequencies with which these alleles occur in that population are 0.06, 0.03, 0.84, 0.03, and 0.04, respectively (Parvanov et al. 2010). Assume that in this population, the two alleles present in any individual are independently sampled from the population as a whole (this can happen if people in the community marry and produce children randomly with respect to Prdm9 genotype).

a. What is the probability that a single allele chosen at random from this population is either A₁ or A₄? 

b. What is the probability that an individual has two A₁ (i.e., what  is the probability that its first allele is A₁ and its second allele is also A₁)?

c. What is the probability that an individual has one A₁ allele and one A₃  allele? (Note that this can happen if the first allele drawn is A₁  and the second is A₃ or if the first allele is A₃ and the second is A₁.  probability tree will help to keep track of all the possibilities.)

d. What is the probability that an individual is not A₁ (i.e., does not have two A₁ alleles)?

e. What is the probability, if you drew two individuals at random from this population, that neither of them would have anA₁A₁ genotype?

f. What is the probability, if you drew two individuals at random from this population, that at least one of them would have an A₁,A₁genotype?

g. What is the probability that three randomly chosen individuals would have no A₂ or A₃ alleles? (Remember that each individual has two alleles.)