Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A nationwide survey conducted by the National Cancer Society revealed the following information. Of 10,000 people surveyed, 3200 were "heavy coffee drinkers" and 160 had cancer of the pancreas. Of those who had cancer of the pancreas, 132 were heavy coffee drinkers. Using the data in this survey, determine whether the events "being a heavy coffee drinker" and "having cancer of the pancreas" are independent events.

Step-by-Step Solution

Answer

The events "being a heavy coffee drinker" and "having cancer of the pancreas" are not independent events, as the calculated probabilities P(A and B) = 132/10000 ≠ P(A) * P(B) = 512000/100000000.

Step 1: Define the probabilities of the events

First, let's define the probabilities of the events: - Event A: being a heavy coffee drinker; - Event B: having cancer of the pancreas. We can calculate the probabilities of these events using the following formulas: - P(A) = the number of heavy coffee drinkers / the total number of people surveyed; - P(B) = the number of people with cancer of the pancreas / the total number of people surveyed. Using the data from the survey, we have: - P(A) = 3200 / 10000; - P(B) = 160 / 10000.

Step 2: Check if the events are independent

Now we can check if the events A and B are independent. To do this, we need to calculate the probability of both events occurring, P(A and B), and check if it's equal to the product of the individual probabilities, P(A) * P(B). The probability of both events occurring can be calculated as follows: - P(A and B) = number of heavy coffee drinkers with cancer of the pancreas / the total number of people surveyed = 132 / 10000. Now, let's check if P(A and B) = P(A) * P(B). - P(A and B) = 132 / 10000; - P(A) * P(B) = (3200 / 10000) * (160 / 10000) = 512000 / 100000000. To determine if the events are independent, we compare P(A and B) and P(A) * P(B): - 132 / 10000 = 512000 / 100000000. Since these probabilities are not equal, the events "being a heavy coffee drinker" and "having cancer of the pancreas" are not independent events.

Key Concepts

Understanding Probability

Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty.

When calculating probabilities, it's crucial to understand the context of the problem. For example, if a nationwide survey indicates that out of 10,000 individuals, 3,200 qualify as heavy coffee drinkers, the probability of a randomly selected person being a heavy coffee drinker (\( P(A) \) is calculated by dividing the number of heavy coffee drinkers by the total number of people surveyed (\( P(A) = \frac{3200}{10000} \) or 0.32, which means there's a 32% chance for a randomly chosen individual to be a heavy coffee drinker.

In the same vein, if 160 people have pancreatic cancer out of the 10,000 surveyed, the probability of a person having cancer of the pancreas (\( P(B) \) is \( P(B) = \frac{160}{10000} \) or 0.016. Understanding how these probabilities are calculated is essential for analyzing the events' relationship and making informed conclusions about their independence or dependence.

Statistical Independence of Events

Statistical independence is a key concept when examining the relationship between two events. Two events, A and B, are independent if the occurrence of one event has no effect on the occurrence of the other. To put it another way, events A and B are independent if the probability of A occurring is the same whether B occurs or not, and vice versa.

To test for independence, mathematicians use a simple formula: if the probability of both events occurring together (\( P(A \text{ and } B) \) equals the product of their individual probabilities (\( P(A) \times P(B) \) then the events are deemed independent. If the results differ, then the events are dependent.

For instance, in the context of the nationwide survey, we calculate \( P(A \text{ and } B) \) by identifying individuals who are both heavy coffee drinkers and have pancreatic cancer, 132 out of 10,000, which yields \( P(A \text{ and } B) = \frac{132}{10000} \) or 0.0132. When comparing this to the product of \( P(A) \times P(B) \) (which is a different value), it becomes evident the events - 'being a heavy coffee drinker' and 'having cancer of the pancreas' are dependent, as their occurrence is related rather than purely coincidental.

The Relationship Between Pancreatic Cancer and Coffee Consumption

The link between lifestyle factors, like coffee consumption, and health conditions such as pancreatic cancer is a topic of exploration in many epidemiological studies. However, interpreting such data requires careful consideration as correlation does not necessarily imply causation.

For example, the survey data showed 132 individuals who are both heavy coffee drinkers and have pancreatic cancer, which may initially suggest a connection. Yet, this observation alone is insufficient to establish a causal relationship. Additional studies would need to consider various confounding factors, adjust for them accordingly, and delve deeper into understanding the biological mechanisms that could potentially link coffee consumption to cancer risks.

It is essential to understand that a statistical association, like the one observed, does not prove that drinking coffee causes pancreatic cancer. Comprehensive research is necessary to understand these complex interactions fully. For educational purposes and for those studying probabilities and statistics, questions such as these highlight the importance of not jumping to conclusions based on limited data and showcase the need for more robust scientific evidence to substantiate such claims.