Gregor Mendel famously carried out experiments on sweet peas. Three naturally occurring varieties of sweet peas have purple, red and white flowers. The flower colour may be assumed to be determined by two genes. The first gene has two alleles: R and r. A sweet pea with one or more R alleles will have red flowers. The second gene also has two alleles: P and p. A sweet pea with no R alleles but with one or more P alleles will have purple flowers. If the sweet pea's alleles are rr and pp, the flowers will be white. Let x denote the proportion of R alleles for the first gene amongst the all the sweet peas cultivated in a particular garden centre, with 1 x the proportion of r alleles; let y similarly represent the proportion of P alleles, 1 y the proportion of p alleles.

1. Calculate, in terms of x and y, the probability that a randomly selected sweet pea from the garden centre has (i) purple flowers, (ii) red flowers, (iii) white flowers.

2. Given that a = 48.16% of the sweet peas in the garden centre have red flowers and b = 48.06% have purple flowers, calculate the values of x and y. (The values of a and b will be provided separately.)

3. A sweet pea with purple flowers is selected at random from the garden centre. What is the probability that it has two P alleles?

4. The pollen from a sweet pea that has two P alleles always carries the P allele, whereas the pollen from a sweet pea that has one P and one p allele has a 50/50 chance of carrying either allele. A species of bee visits only purple-flowered sweet peas: seeds that develop from pollination by such a bee carry one allele from the pollen and one from the plant which was pollinated. What is the probability that a seed of this type carries two P alleles?