Problem 128 - Applied Statistics and Probability for Engineers

Question

Heart failures are due to either natural occurrences \((87 \%)\) or outside factors \((13 \%) .\) Outside factors are related to induced substances \((73 \%)\) or foreign objects \((27 \%) .\) Natural occurrences are caused by arterial blockage \((56 \%),\) disease \((27 \%),\) and infection (e.g., staph infection) (17\%). (a) Determine the probability that a failure is due to an induced substance (b) Determine the probability that a failure is due to disease or infection.

Step-by-Step Solution

Answer

(a) 9.49%; (b) 38.28%.

Step 1: Understanding the Problem

We have two types of heart failures: natural occurrences and outside factors with their respective probabilities. Additionally, each category is further divided into subcategories with given probabilities. We need to determine the probability of failure due to specific causes.

Step 2: Probability of Failure from Induced Substance

To find the probability of a heart failure due to induced substances, we consider that outside factors, which contribute 13% to heart failures, are related to induced substances 73% of the time. We calculate this probability by multiplying the probabilities of these two events: \( P(\text{Induced Substance}) = P(\text{Outside Factors}) \times P(\text{Induced Substance} \,|\, \text{Outside Factors}) \). So, \( P(\text{Induced Substance}) = 0.13 \times 0.73 = 0.0949 \) or 9.49%.

Step 3: Probability of Failure from Disease or Infection

For the probability of a heart failure due to disease or infection, we need to consider natural occurrences, which contribute 87% to heart failures. Disease accounts for 27% of natural causes and infection accounts for 17% of natural causes. We sum the probabilities of these events after accounting for their shared category: \( P(\text{Disease or Infection}) = P(\text{Disease} \,|\, \text{Natural}) + P(\text{Infection} \,|\, \text{Natural}) \). Therefore, \( P(\text{Disease or Infection}) = 0.27 + 0.17 = 0.44 \), and then multiply by the probability of natural occurrences: \( P(\text{Disease or Infection}) = 0.87 \times 0.44 = 0.3828 \) or 38.28%.

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in statistics that allows us to determine the likelihood of an event occurring, given that another related event has occurred. In simple terms, it helps us calculate the chances of something happening under certain conditions. When we calculated the probability of heart failure due to an induced substance, we used conditional probability. We knew the probability that heart failures were caused by outside factors, and we knew the percentage of those due to induced substances. By multiplying these probabilities, we determined the likelihood of heart failures specifically due to induced substances.

The formula for conditional probability is generally written as follows:

• \( P(A \,|\, B) \), which reads "the probability of A given B," is calculated by \( P(A \cap B) / P(B) \).

This allows us to focus on a specific subset of our sample data, pinpointing the exact conditions we need to analyze.

Understanding conditional probability provides clarity in complex scenarios, such as the multiple causes of heart failure in our problem. It's like zooming in on a particular layer of data, helping us make more precise predictions.

Probability Distribution

Probability distribution is a way to describe how the probabilities of different outcomes of a random event are spread out. It shows us what values the random variable can take and how likely each value is.

In our exercise, we see a form of discrete probability distribution. Heart failures are divided into specific categories like natural occurrences and outside factors, as well as their subcategories. Each category and subcategory has an associated probability that adds up to 1, representing the total possible outcomes. For example:

• Heart failures due to natural occurrences account for 87% of cases.
• Those due to outside factors are 13%.
• Further subdivisions categorize causes of failures like artery blockage or induced substances.

These probabilities follow the rules of a distribution, where the sum of all probabilities within a particular classification equals 1, ensuring that all possible outcomes are covered.

The beauty of probability distributions lies in their ability to help us understand complex data by organizing it into understandable segments. By visualizing or computing these distributions, we can predict future scenarios and make informed decisions based on statistical analysis.

Statistical Analysis

Statistical analysis involves examining and interpreting data to uncover patterns and trends. It encompasses a variety of methods that help us make sense of numerical data and guide future decisions or predictions.

In our heart failure problem, statistical analysis allowed us to break down the overall data into more manageable parts. By calculating each probability and analyzing the data, we could determine how likely certain causes of heart failure were. This process involved:

• Collecting data on heart failures and their causes.
• Applying probability techniques to analyze the data.
• Drawing conclusions about which factors are most influential.

Through statistical analysis, we uncover deeper insights into the phenomena we are studying. This enables us to view the big picture while also addressing specific questions like the likelihood of heart failures arising from disease or infection.

In practice, statistical analysis helps industries and sciences improve outcomes, making it a powerful tool in decision-making processes.