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Chapter 3: Problem 3

A research study uses 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 are selected randomly and tested for the marker, what is the probability that exactly 1 man has the marker? b. If 10 men are selected randomly and tested for the marker, what is the probability that more than 1 has the marker?

Short Answer

Expert verified
a. Probability of exactly 1 man with the marker: 0.121. b. Probability of more than 1 man: 0.851.

Step by step solution

01

Define the Problem and Variables

We need to find the probabilities related to selecting 10 men from a group where 30% carry a genetic marker. This situation can be modeled using a binomial distribution. Define: \( n = 10 \) (the number of trials), \( p = 0.3 \) (the probability of success鈥攈aving the marker), and the number of successes \( X \) we are interested in.
02

Probability of Exactly 1 Success

The probability of exactly \( k = 1 \) success (marker present) in a binomial distribution is calculated using the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substitute \( n = 10 \), \( k = 1 \), and \( p = 0.3 \): \[ P(X = 1) = \binom{10}{1} (0.3)^1 (0.7)^9 \] Calculate this to find the probability.
03

Calculate for Exactly 1 Success

Calculate \( \binom{10}{1} = 10 \), and then the probability: \[ P(X = 1) = 10 \times 0.3 \times (0.7)^9 \] \[ = 10 \times 0.3 \times 0.0403536067 \]\[ \approx 0.121060821 \] Thus, the probability that exactly 1 man has the marker is approximately 0.121.
04

Probability of More Than 1 Success

"More than 1 success" means \( X \geq 2 \). First, calculate the probabilities for 0 and 1 success and subtract from 1: \[ P(X \geq 2) = 1 - (P(X = 0) + P(X = 1)) \].
05

Calculate for Zero Success

Using the binomial probability formula, calculate for no success: \[ P(X = 0) = \binom{10}{0} (0.3)^0 (0.7)^{10} \] \[ = 1 \times 1 \times (0.7)^{10} \] \[ \approx 0.0282475249 \].
06

Calculate for More Than 1 Success

Use the results from Step 3 and Step 5 to find: \[ P(X \geq 2) = 1 - (0.0282475249 + 0.121060821) \] \[ = 1 - 0.1493083459 \] \[ = 0.8506916541 \]. Thus, the probability that more than 1 man has the marker is approximately 0.851.

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics concerned with analyzing random events and quantifying uncertain outcomes. It provides tools to reason about the likelihood of different outcomes in a systematic way. In this context, we are interested in probabilistic events involving genetic markers.
  • Probability ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
  • A key concept in probability is the idea of a random variable, which is a variable that takes on different values due to randomness.
  • Events can be independent (the outcome of one event does not affect the outcome of another) or dependent.
In our exercise, identifying how many men carry a genetic marker involves calculating the probability of certain outcomes (such as exactly 1 or more than 1 man having the marker) using probability theory.
Statistical Modeling
Statistical modeling is a powerful tool to represent real-world situations mathematically. In our exercise, we use the binomial distribution as a model. This is an example of a discrete probability distribution.
  • Statistical modeling helps to simplify complex systems, making calculations and predictions feasible.
  • The binomial distribution is suitable for scenarios with two possible outcomes (such as having or not having a marker).
The binomial model assumes a fixed number of identical trials, each with the same probability of success. For instance, when testing 10 men, the model assumes each has a 30% chance of carrying the marker, independently of the others. This simplification allows for the calculation of probabilities using binomial formulas.
Genetic Markers
Genetic markers are specific DNA sequences that help in identifying particular genetic conditions. They can be linked to health risks, such as high blood pressure, which is the focus of our study.
  • Markers are often used in genetic research to assess risks associated with certain conditions.
  • In research, these markers are crucial for identifying at-risk individuals.
In the context of probability and statistical modeling, genetic markers are a type of biological variable that can be analyzed to understand the likelihood of disease occurrence in a population. For example, determining what percentage of men carry a risk-enhancing marker provides insight into population health and helps in targeted health interventions.
Risk Assessment
Risk assessment is a process used to identify and evaluate risks posed by certain factors, like genetic markers, to individuals or populations. It's an essential part of public health and healthcare planning.
  • It involves estimating the probability that a particular adverse outcome will occur, given certain conditions or factors.
  • Risk assessment in genetics involves understanding the linkage between genetic markers and diseases.
By calculating the probability of a genetic marker presence among a sample, researchers can assess the potential health risks. In our example, knowing the probability of men carrying a high blood pressure marker aids in predicting how this condition might affect a population. Such assessments allow for better preparedness and healthcare resource allocation.

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