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Chapter 3: Problem 145

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 is approximately 0.12105. (b) Probability is approximately 0.8507.

Step by step solution

01

Identify the random variable and distribution

The problem involves determining the probability of a certain number of successes out of a fixed number of trials. Here, the random variable is the number of men with the marker out of 10 chosen. The distribution type that best fits this problem is the binomial distribution because each man has a 30% (0.3) probability of having the marker, independently of others. The number of trials, n, is 10, and the probability of success, p, is 0.3.
02

Use the binomial probability formula for exactly 1 success

The formula for the probability of getting exactly k successes in n binomial trials is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). For the probability that exactly 1 man out of the 10 has the marker, set \( n = 10 \), \( k = 1 \), and \( p = 0.3 \). Calculate:\[ P(X = 1) = \binom{10}{1} (0.3)^1 (0.7)^9 \].
03

Calculate \( P(X = 1) \)

Compute the binomial coefficient and each term in the expression:- \( \binom{10}{1} = 10 \)- \( (0.3)^1 = 0.3 \)- \( (0.7)^9 \approx 0.04035 \)Multiply these values together:\[ P(X = 1) = 10 \times 0.3 \times 0.04035 = 0.12105 \].
04

Use the binomial probability formula for more than 1 success

To find the probability that more than 1 man has the marker, calculate the complementary probabilities for 0 and 1, then subtract these from 1:\[ P(X > 1) = 1 - (P(X = 0) + P(X = 1)) \].
05

Calculate \( P(X = 0) \)

Use the binomial probability formula:\[ P(X = 0) = \binom{10}{0} (0.3)^0 (0.7)^{10} \].- \( \binom{10}{0} = 1 \)- \( (0.3)^0 = 1 \)- \( (0.7)^{10} \approx 0.0282475249 \)Thus, \( P(X = 0) = 1 \times 1 \times 0.0282475249 = 0.02825 \).
06

Calculate \( P(X > 1) \)

Now calculate the probability of more than one marker:- From Step 3, \( P(X = 1) = 0.12105 \)- From Step 5, \( P(X = 0) = 0.02825 \)Thus, \[ P(X > 1) = 1 - (0.02825 + 0.12105) = 1 - 0.1493 = 0.8507 \].

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Key Concepts

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

Probability
Probability is a measure of the likelihood that a particular event will occur. In the scenario of selecting men with a genetic marker, probability helps us understand how likely it is to select a certain number of men from a group who carry this marker. In our problem, we apply probability principles to determine outcomes when ten men are randomly chosen. Each man has a 30% chance of carrying the marker, independent of the others. Here's a simple breakdown of what probability represents:
  • It is a number between 0 and 1.
  • Probability of 0 indicates an impossible event.
  • Probability of 1 means the event is certain.
Thus, understanding probability allows us to calculate and predict the likelihood of different scenarios, such as all ten men having the marker, none having it, or some possessing it. By using mathematical formulas, we can distinguish these possibilities.
Random Variable
A random variable is an essential concept in probability and statistics. It is a variable whose possible values are numerical outcomes of a random phenomenon. In our exercise, the random variable represents the number of men with the marker. This number can range from 0 to 10, depending on how many out of the ten randomly selected men actually carry the genetic marker. Types of random variables include:
  • Discrete random variable: Takes on a countable number of distinct values. The variable in our example is discrete as it counts the number of men with the marker.
  • Continuous random variable: Can take on an infinite number of possible values. These typically represent measurements.
The random variable simplifies the description of outcomes to numbers that can be analyzed using probability distributions. In this binomial scenario, it makes calculations manageable by quantitatively expressing outcomes.
Binomial Probability Formula
The binomial probability formula is crucial for calculating the likelihood of achieving a fixed number of successes in a set number of independent trials.Let's dive into the formula specifics:The probability of exactly \( k \) successes in \( n \) trials can be found using \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where:
  • \( \binom{n}{k} \) is the binomial coefficient, which gives the number of ways to pick \( k \) successes from \( n \) trials.
  • \( p \) is the probability of a success on a single trial.
  • \( (1-p) \) is the probability of a failure on a single trial.
In our study about genetic markers, the binomial formula helps us compute probabilities of various outcomes like exactly one man having the marker or more than one having it. Thus, the formula offers a systematic approach to tackle probability problems with specific conditions, efficiently yielding probabilities.

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