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Chapter 3: Problem 121

A company employs 800 men under the age of 55 . Suppose that \(30 \%\) carry a marker on the male chromosome that indicates an increased risk for high blood pressure. (a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly one man has the marker? (b) If 10 men in the company are tested for the marker in this chromosome, what is the probability that more than one has the marker?

Short Answer

Expert verified
(a) Approximately 0.1211. (b) Approximately 0.8507.

Step by step solution

01

Define the Problem

We have two probabilities to calculate related to the presence of a marker on the male chromosome. We are given that 30% of the men have this marker, and we need to find probabilities related to tests conducted on 10 men.
02

Set Up a Binomial Distribution

This problem can be modeled using a binomial distribution, where the number of trials is 10 (since 10 men are tested), and the probability of success (finding a man with the marker) is 0.3 (30%).
03

Calculate Probability For Exactly One Success

The probability of exactly one man having the marker is given by the binomial probability formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k},\]where \( n = 10\), \( k = 1\), and \( p = 0.3 \). Therefore:\[P(X = 1) = \binom{10}{1} (0.3)^1 (0.7)^9.\]
04

Calculate Binomial Coefficient and Probability

First, calculate the binomial coefficient \( \binom{10}{1} = 10 \). Then, calculate the probability:\[P(X = 1) = 10 \cdot 0.3 \cdot 0.7^9 \approx 0.1211.\]
05

Calculate Probability For More Than One Success

To find the probability of more than one man having the marker, we need to calculate:\[P(X > 1) = 1 - P(X \leq 1) = 1 - (P(X = 0) + P(X = 1)).\]
06

Calculate Probability for Zero Successes

Calculate \( P(X = 0) \):\[P(X = 0) = \binom{10}{0} (0.3)^0 (0.7)^{10} = 1 \cdot 0.7^{10} \approx 0.0282.\]
07

Calculate Probability of More Than One Success

Now calculate:\[P(X > 1) = 1 - (0.0282 + 0.1211) = 1 - 0.1493 \approx 0.8507.\]
08

Record the Results

For part (a), the probability that exactly one man has the marker is approximately 0.1211. For part (b), the probability that more than one man has the marker is approximately 0.8507.

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Key Concepts

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

Probability Calculation
Probability calculation is a key part of understanding scenarios where outcomes are random. In this case, calculating the probability helps determine how likely it is for a certain number of men to have a chromosome marker. Probability ranges from 0 to 1, where 0 means an event will not happen, and 1 means it definitely will. In our scenario:
  • We're interested in two specific probabilities: the probability of exactly one man having the marker and the probability of more than one man having it.
  • This involves using the Binomial Distribution, as we deal with a fixed number of trials (10 men), each with two possible outcomes (having the marker or not).
To find the probability of exactly one success, the binomial probability formula is utilized.
The chance of a single man having the marker is 0.3, while not having it is 0.7. By using these probabilities, you can compute the probability of such events using formulas derived from the binomial distribution.
Chromosome Marker
A chromosome marker is a genetic indicator used in this context to reflect an increased risk for a health condition, such as high blood pressure. Here, 30% of the men are known to carry this marker on their male chromosome.
  • The presence of the chromosome marker is akin to finding specific genetic traits that might influence health risks.
  • Markers can provide vital information for predicting health patterns within a population.
Understanding how many people might carry such markers can help organizations make informed decisions about health programs and interventions.
In practical terms, identifying how many men out of a sample carry this marker can also aid in planning medical research or monitoring public health.
Binomial Coefficient
The binomial coefficient is a fundamental part of the binomial probability formula used to calculate chances of certain outcomes when looking at trials. In the formula \( \binom{n}{k} \), \( n \) is the total number of trials, and \( k \) is the number of successful trials we are interested in.
  • It represents the number of different ways you can choose \( k \) successes out of \( n \) total trials.
  • In our scenario, with 10 men being tested and wanting to know the probability for exactly one carrying the marker, \( \binom{10}{1} \) tells us how many different ways we can choose one man out of 10 to carry the marker.
The coefficient, combined with the probabilities for success and failure, forms the backbone of the probability formula we used to find out specific probabilities related to the chromosome marker presence.
Understanding and calculating the binomial coefficient is crucial as it helps translate the concept of random sampling into tangible probabilities.

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Most popular questions from this chapter

In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1 . (a) What is the probability four or more people will have to be tested before two with the gene are detected? (b) How many people are expected to be tested before two with the gene are detected?

In 1898 L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of 0.61 . (a) What is the probability of more than one death in a corps in a year? (b) What is the probability of no deaths in a corps over five years?

From 500 customers, a major appliance manufacturer will randomly select a sample without replacement. The company estimates that \(25 \%\) of the customers will provide useful data. If this estimate is correct, what is the probability mass function of the number of customers that will provide useful data? (a) Assume that the company samples five customers. (b) Assume that the company samples 10 customers.

This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, \(2 \%\) of the components are identified as defective, and the components can be assumed to be independent. (a) If the manufacturer stocks 100 components, what is the probability that the 100 orders can be filled without reordering components? (b) If the manufacturer stocks 102 components, what is the probability that the 100 orders can be filled without reordering components? (c) If the manufacturer stocks 105 components, what is the probability that the 100 orders can be filled without reordering components?

Determine the cumulative distribution function of a binomial random variable with \(n=3\) and \(p=1 / 2\).
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