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.
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 6.23% of the sweet peas in the garden centre have red flowers and 5.95% have purple flowers, calculate the values of x and y.
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?
1. To calculate the probability of the different flower colors, we need to consider the possible combinations of alleles for each color.
(i) Purple flowers: A sweet pea can have purple flowers if it has no R alleles but has one or more P alleles (rr and P_ or rr and PP). Since the second gene has two alleles (P and p), we can represent the probability of having a P allele as x. Therefore, the probability of a randomly selected sweet pea having purple flowers is x.
(ii) Red flowers: A sweet pea can have red flowers if it has one or more R alleles, regardless of the P allele (RR, Rr and PP, Rr and Pp). The probability of having an R allele is 1 - x, as we have already considered the probability of having a P allele for purple flowers. Therefore, the probability of a randomly selected sweet pea having red flowers is 1 - x.
(iii) White flowers: A sweet pea can have white flowers if it has both rr and pp alleles. Since both genes are independent, the probability of having rr is (1 - x) and the probability of having pp is (1 - y), where y represents the probability of having a P allele. Therefore, the probability of a randomly selected sweet pea having white flowers is (1 - x) * (1 - y).
2. With the given information, we can set up an equation system:
Let R% represent the percentage of sweet peas with red flowers and P% represent the percentage of sweet peas with purple flowers.
R% = (1 - x) * (1 - y)
P% = x
We are given that R% = 6.23% and P% = 5.95%. Substituting these values into the equations:
6.23% = (1 - x) * (1 - y)
5.95% = x
From the second equation, x = 0.0595.
Substituting x = 0.0595 into the first equation:
6.23% = (1 - 0.0595) * (1 - y)
0.0623 = 0.9405 * (1 - y)
y = 1 - (0.0623 / 0.9405)
y ≈ 0.9334
Therefore, x ≈ 0.0595 and y ≈ 0.9334.
3. The probability that a sweet pea with purple flowers has two P alleles can be found by considering the given information that purple flowers can be obtained with both PP and Pp combinations.
The probability of having purple flowers with two P alleles (PP) is x, as we defined earlier. The probability of having purple flowers with one P allele (Pp) is (1 - x). Since these two possibilities are mutually exclusive, the sum of their probabilities gives the overall probability of purple flowers.
Therefore, the probability that a sweet pea with purple flowers has two P alleles is x.
4. To calculate the probability that a seed carries two P alleles, we need to consider the possible combinations of alleles from both the pollen and the plant.
The pollen from a sweet pea with two P alleles always carries the P allele, so the probability is 1.
For a sweet pea with one P allele and one p allele, there is a 50/50 chance of carrying either allele. Therefore, the probability of carrying a P allele from this combination is 0.5.
Since the species of bee only visits purple-flowered sweet peas, we only need to consider the purple flower combinations.
The probability that a seed carries two P alleles is the product of the probabilities of each combination:
Probability = (x * 1) + ((1 - x) * 0.5)
This simplifies to:
Probability = x + (0.5 - 0.5x)
Therefore, the probability that a seed from this type of pollination carries two P alleles is x + (0.5 - 0.5x).