Question

In Exercises 5–36, express all probabilities as fractions.

Designing Experiment Clinical trials of Nasonex involved a group given placebos and another group given treatments of Nasonex. Assume that a preliminary phase I trial is to be conducted with 12 subjects, including 6 men and 6 women. If 6 of the 12 subjects are randomly selected for the treatment group, find the probability of getting 6 subjects of the same gender. Would there be a problem with having members of the treatment group all of the same gender?

Short answer

The probability of selecting six subjects of the same gender is$\frac{1}{462}$.

Drugs can initiate different reactions in males and females. Thus, if the treatment group contains all males/all females, the effectiveness of the drug cannot be identified accurately, and the drug cannot be used in general for all genders.

Step-by-step solution

Given information

The total number of subjects to be chosen from 12 subjects is six.

Out of the 12 subjects, six are males, and six are females

Define combination

In combination, the order of the selections of say r units from n is not important. The formula for this is shown below:

${}^n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Compute the number of ways of selecting the same gender

The genders are male and female.

Case 1: Selecting all male subjects

The total number of subjects is 12.

The number of subjects to be chosen is six.

The number of ways in which six subjects can be chosen from 12 (in any order) is

$$\begin{gathered} {}^{12}{C}_6=\frac{12!}{\left(12-6\right)!\times 6!} \\ =924 \end{gathered}$$

The number of ways in which all male subjects can be selected is one.

The probability of selecting all male subjects is

$P\left(\text{all}\text{male}\right)=\frac{1}{924}.$

Case 2: Selecting all female subjects

The number of ways in which all female subjects can be chosen is one.

The probability of selecting all female subjects is

$P\left(\text{all}\text{female}\right)=\frac{1}{924}$

Compute the probability of selecting either gender

Let A be the event of selecting all subjects of the same gender.

The probability of selecting all subjects of the same gender is the sum of the probabilities of selecting all male subjects and all female subjects. The calculation for this is shown below:

$$\begin{gathered} P\left(A\right)=\frac{1}{924}+\frac{1}{924} \\ =\frac{1}{462} \end{gathered}$$

Therefore, the probability of selecting subjects of the same gender is$\frac{1}{462}$.

State the potential issue

The drug test is more reliable if handled randomly over all subjects and not specifically over a gender.

As drugs can react differently in males and females, the correct conclusion cannot be made on the effectiveness of the treatment if the treatment group has subjects of the same gender.