Question

Critical Thinking: Did Mendel’s results from plant hybridization experiments contradict his theory? Gregor Mendel conducted original experiments to study the genetic traits of pea plants. In 1865 he wrote “Experiments in Plant Hybridization,” which was published in Proceedings of the Natural History Society. Mendel presented a theory that when there are two inheritable traits, one of them will be dominant and the other will be recessive. Each parent contributes one gene to an offspring and, depending on the combination of genes, that offspring could inherit the dominant trait or the recessive trait. Mendel conducted an experiment using pea plants. The pods of pea plants can be green or yellow. When one pea carrying a dominant green gene and a recessive yellow gene is crossed with another pea carrying the same green>yellow genes, the offspring can inherit any one of these four combinations of genes: (1) green/green; (2) green/yellow; (3) yellow/green; (4) yellow/yellow. Because green is dominant and yellow is recessive, the offspring pod will be green if either of the two inherited genes is green. The offspring can have a yellow pod only if it inherits the yellow gene from each of the two parents. Given these conditions, we expect that 3/4 of the o§spring peas should have green pods; that is, P(green pod) = 3/4. When Mendel conducted his famous hybridization experiments using parent pea plants with the green/yellow combination of genes, he obtained 580 offspring. According to Mendel’s theory, 3/4 of the offspring should have green pods, but the actual number of plants with green pods was 428. So the proportion of offspring with green pods to the total number of offspring is 428/580 = 0.738. Mendel expected a proportion of 3/4 or 0.75, but his actual result is a proportion of 0.738.

a. Assuming that P(green pod) = 3/4, find the probability that among 580 offspring, the number of peas with green pods is exactly 428.

b. Assuming that P(green pod) = 3/4, find the probability that among 580 offspring, the number of peas with green pods is 428 or fewer.

c. Which of the two preceding probabilities should be used for determining whether 428 is a significantly low number of peas with green pods?

d. Use probabilities to determine whether 428 peas with green pods is a significantly low number. (Hint: See “Identifying Significant Results with Probabilities” in Section 5-1.)

Short answer

a. The probability of selecting exactly 428 peas with green pods is equal to 0.0301.

b.The probability of selecting 428 or fewer peas with green pods is equal to 0.265.

c. Since the probability of 428 or fewer peas with green pods is represented in part b, theprobability computed in part b is relevant for determining whether the result of green peas is significantly low.

d. Since the probability of 428 or fewer green peas is not less than or equal to 0.05, the value of 428 green peas is not significantly low.

Step-by-step solution

Given information

It is given that the probability of a green pod is equal to ¾.

Binomial probability

a.

Let X denote the number of peas with green pods.

Success is defined as selecting a pea with a green pod.

The probability of success is equal to as follows:

$$\begin{gathered} p=\frac{3}{4} \\ =0.75 \end{gathered}$$

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.75 \\ =0.25 \end{gathered}$$

The number of trials (n) is equal to 580.

The binomial probability formula used to compute the given probability is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

Using the binomial probability formula, the probability of selecting 428 peas with green pods is given as follows:

$$\begin{gathered} P\left(X=428\right)={}^{580}{C}_{428}{\left(0.75\right)}^{428}{\left(0.25\right)}^{580-428} \\ =0.0301 \end{gathered}$$

Thus, the probability of selecting exactly 428 peas with green pods is equal to 0.0301.

b.

The probability of 428 or fewer peas with green pods has the following expression:

$$\begin{gathered} P\left(X⩽428\right)=1-P\left(X>428\right) \\ =1-\left[P\left(X=429\right)+P\left(X=430\right)+.......+P\left(X=580\right)\right] \end{gathered}$$

The individual probabilities will be computed as follows:

$$\begin{gathered} P\left(X=429\right)={}^{580}{C}_{429}{\left(0.75\right)}^{429}{\left(0.25\right)}^{151} \\ =0.032019 \\ P\left(X=430\right)={}^{580}{C}_{430}{\left(0.75\right)}^{430}{\left(0.25\right)}^{150} \\ =0.033732 \\ . \\ . \\ . \\ P\left(X=580\right)={}^{580}{C}_{580}{\left(0.75\right)}^{580}{\left(0.25\right)}^0 \\ \approx 3.432\times {10}^{-73} \end{gathered}$$

Thus, the required probability is computed as follows:

$$\begin{gathered} P\left(X⩽428\right)=1-P\left(X>428\right) \\ =1-\left[P\left(X=429\right)+P\left(X=430\right)+.......+P\left(X=580\right)\right] \\ =1-0.735044 \\ =0.264956 \\ \approx 0.265 \end{gathered}$$

Thus, the probability of selecting 428 or fewer peas with green pods is equal to 0.265.

Examining the significance of a value using the probability formula

c.

The probability formula to determine whether the given value of the number of successes (x) is significantly low or not is shown below:

$P\left(x\text{or}\text{fewer}\right)⩽0.05$

Here, the value of x considered is equal to 428.

Since the probability of 428 or fewer peas with green pods is represented in part b, it can be concluded that theprobability computed in part b is relevant for determining whether the result of green peas is significantly low.

d.

$$\begin{gathered} P\left(428\text{or}\text{fewer}\right)=0.265 \\ >0.05 \end{gathered}$$

Since the probability of 428 or fewer green peas is not less than or equal to 0.05, the value of 428 green peas is not significantly low.