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Chapter 4: Problem 30

Express all probabilities as fractions. 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

Expert verified
The probability is \( \frac{1}{462} \). Having all members of the same gender would be problematic.

Step by step solution

01

- Understand the Problem

The problem involves randomly selecting 6 subjects out of 12 for a clinical trial group and finding the probability that all selected subjects are of the same gender.
02

- Identify Total Number of Ways to Select 6 Subjects

Calculate the total number of ways to randomly select 6 subjects out of 12. This can be done using the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, n = 12 and k = 6.\[ \binom{12}{6} = \frac{12!}{6!6!} = 924 \]
03

- Calculate the Number of Favorable Outcomes for All Men

Calculate the number of ways to select all 6 subjects as men from the 6 available men.\[ \binom{6}{6} = 1 \]
04

- Calculate the Number of Favorable Outcomes for All Women

Calculate the number of ways to select all 6 subjects as women from the 6 available women.\[ \binom{6}{6} = 1 \]
05

- Calculate Total Number of Favorable Outcomes

Add the number of favorable outcomes for all men and all women.Number of favorable outcomes = 1 (all men) + 1 (all women) = 2
06

- Calculate the Probability

Find the probability by dividing the total number of favorable outcomes by the total number of possible outcomes.\[ P(\text{6 of the same gender}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{2}{924} = \frac{1}{462} \]
07

- Interpret the Result

Having all members of the treatment group of the same gender would be problematic because it could lead to biased results, not accounting for potential gender differences in response to the treatment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

random selection
Random selection is a fundamental concept in probability and statistics, critical to clinical trials. In random selection, every subject has an equal chance of being chosen. This helps to ensure that the trial results are unbiased and representative of the larger population. In our exercise, we need to select 6 subjects out of 12 for a treatment group. Since the selection is random, each of the 12 subjects has an equal chance of being chosen. This reduces the risk of systematic bias and provides a fair representation of outcomes across different groups of subjects.
combinatorics
Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting. It helps determine the number of ways to select items from a larger set, where the order of selection does not matter. In our problem, we use combinations since we are selecting 6 subjects out of 12 without regard to the order. The combination formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is used, where \( n \) is the total number of items, and \( k \) is the number of items to choose. Applying this, we calculate \( \binom{12}{6} = 924 \), showing there are 924 possible ways to pick 6 out of 12 subjects.
gender bias in studies
Gender bias in clinical studies occurs when there is an unequal representation of genders in treatment or control groups, leading to skewed results. In our exercise, if all 6 subjects selected for treatment are of the same gender, the trial results may not accurately reflect how both men and women respond to the treatment, thus causing gender bias. This is why having a mix of genders in both the control and treatment groups is essential. Equal representation ensures that the treatment's effectiveness and side effects are fairly evaluated for all genders, leading to more reliable and generalizable results.
clinical trial design
Clinical trial design involves planning and structuring a study to ensure reliable and valid results. It includes random selection of subjects, equal representation of demographics, and clear methodologies. In our example, the design requires randomly selecting 6 subjects for a treatment group, considering gender representation to avoid bias. Good trial design ensures that the results are scientifically sound and applicable to the general population. Involving diverse subjects reduces bias and increases the credibility of the findings, providing clear insights into a treatment's efficacy and safety across different population segments.

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Most popular questions from this chapter

Express the indicated degree of likelihood as a probability value between 0 and \(1 .\) When making a random guess for an answer to a multiple choice question on an SAT test, the possible answers are \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\), so there is 1 chance in 5 of being correct.

Find the probability. It has been reported that \(20 \%\) of iPhones manufactured by Foxconn for a product launch did not meet Apple's quality standards. An engineer needs at least one defective iPhone so she can try to identify the problem(s). If she randomly selects 15 iPhones from a very large batch, what is the probability that she will get at least 1 that is defective? Is that probability high enough so that she can be reasonably sure of getting a defect for her work?

A classical probability problem involves a king who wanted to increase the proportion of women by decreeing that after a mother gives birth to a son, she is prohibited from having any more children. The king reasons that some families will have just one boy, whereas other families will have a few girls and one boy, so the proportion of girls will be increased. Is his reasoning correct? Will the proportion of girls increase?

Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Ten percent of people are left-handed. In a study of dexterity, 15 people are randomly selected. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate the random selection of 15 people. Each of the 15 outcomes should be an indication of one of two results: (1) Subject is left- handed; (2) subject is not left-handed.

Use a simulation approach to find the probability that when five consecutive babies are born, there is a run of at least three babies of the same sex. Describe the simulation procedure used, and determine whether such runs are unlikely.
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