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 48.16% of the sweet peas in the garden centre have red flowers and 48.06% 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?
Answer:
1. (i) Purple flowers: xy
(ii) Red flowers: x(1-y)
(iii) White flowers: (1-x)(1-y)
2. x = 0.4816 and y = 0.4806
3. The probability that it has two P alleles is y^2.
4. The probability that a seed of this type carries two P alleles is 0.5 x y^2.
1. To calculate the probabilities, we need to consider the combinations of alleles that lead to each flower color:
(i) Purple flowers: A sweet pea will have purple flowers if it has no R alleles (rr) and at least one P allele (1-y). Therefore, the probability of a randomly selected sweet pea having purple flowers is (1-x)(1-y).
(ii) Red flowers: A sweet pea will have red flowers if it has at least one R allele (x) and any combination of P alleles (y or 1-y). Therefore, the probability of a randomly selected sweet pea having red flowers is x.
(iii) White flowers: A sweet pea will have white flowers if it has no R alleles (rr) and no P alleles (pp). Therefore, the probability of a randomly selected sweet pea having white flowers is (1-x)(1-y).
2. Given that 48.16% of the sweet peas have red flowers (x) and 48.06% have purple flowers (1-y), we can use this information to set up the following equations:
x = 0.4816
1 - y = 0.4806
Solving these equations, we find:
x = 0.4816
y = 1 - 0.4806 = 0.5194
Thus, the values of x and y are 0.4816 and 0.5194, respectively.
3. To find the probability that a sweet pea with purple flowers has two P alleles, we need to consider the possible genotypes for purple flowers. A sweet pea can have purple flowers if it has no R alleles (rr) and any combination of P alleles (y or 1-y).
The probability of having two P alleles can be calculated as the proportion of sweet peas with the genotype pp (no R alleles and both P alleles). So the probability is (1-x)(y).
4. To find the probability that a seed carries two P alleles, we need to consider two scenarios:
(i) Pollen from a sweet pea with two P alleles: The pollen will always carry the P allele (100% probability). The likelihood of this scenario is (y) since it represents the proportion of sweet peas with two P alleles.
(ii) Pollen from a sweet pea with one P and one p allele: The pollen has a 50/50 chance of carrying either allele. The likelihood of this scenario is (1-y) since it represents the proportion of sweet peas with one P and one p allele.
The probability that a seed carries two P alleles can be calculated as the weighted average of these two scenarios. So the probability is (y) + (1-y) * 0.5 = y + 0.5 - 0.5y = 0.5 + 0.5y.
Therefore, the probability that a seed of this type carries two P alleles is 0.5 + 0.5y.
1. To calculate the probability of a randomly selected sweet pea having a certain flower color, we need to consider the possible combinations of alleles for each gene.
(i) Purple flowers: In order for a sweet pea to have purple flowers, it must have no R alleles (rr) and one or more P alleles. From the given information, we know that if the sweet pea has no R alleles but with one or more P alleles, it will have purple flowers. Therefore, the probability of a randomly selected sweet pea having purple flowers is (1 - x) * y.
(ii) Red flowers: A sweet pea with one or more R alleles will have red flowers. Therefore, the probability of a randomly selected sweet pea having red flowers is x.
(iii) White flowers: A sweet pea with rr and pp alleles will have white flowers. The probability of a sweet pea having rr alleles is (1 - x) * (1 - x), and the probability of having pp alleles is (1 - y) * (1 - y). Therefore, the probability of a randomly selected sweet pea having white flowers is (1 - x) * (1 - x) * (1 - y) * (1 - y).
2. The given information states that 48.16% of the sweet peas have red flowers and 48.06% have purple flowers. We can use this information to set up two equations:
x = 0.4816 (probability of red flowers)
(1 - x) * y = 0.4806 (probability of purple flowers)
Solving these equations simultaneously will give us the values of x and y.
3. We are asked to find the probability that a sweet pea with purple flowers has two P alleles. From the information given, we know that if a sweet pea has purple flowers, it must have one or more P alleles. Therefore, we only need to find the probability that a sweet pea has two P alleles given that it has purple flowers. This probability can be represented as y / (y + (1 - y)). Simplifying this expression gives us the probability of having two P alleles as y.
4. Given that the pollen from a sweet pea with two P alleles always carries the P allele, and the pollen from a sweet pea with one P and one p allele has a 50/50 chance of carrying either allele, we can calculate the probability of a seed carrying two P alleles.
If a sweet pea has purple flowers, it must have one or more P alleles. If the sweet pea has two P alleles, the probability that the pollen carries the P allele is 1. If the sweet pea has one P allele and one p allele, the probability that the pollen carries the P allele is 0.5.
Therefore, the probability that a seed carries two P alleles is y (probability of a sweet pea having purple flowers and two P alleles) + (1 - y) * 0.5 (probability of a sweet pea having purple flowers and one P allele and one p allele).