Question

Suppose that \(23 \%\) of adults smoke cigarettes. It's known that \(57 \%\) of smokers and \(13 \%\) of nonsmokers develop a certain lung condition by age 60 . a) Explain how these statistics indicate that lung condition and smoking are not independent. b) What's the probability that a randomly selected 60 -year-old has this lung condition?

Short answer

The events are not independent as smoking affects the probability of lung condition; the probability of the condition is 23.12%.

Step-by-step solution

Understand Independence

Two events, A and B, are independent if the probability of A given B is the same as the probability of A. In this context, the events are 'having a lung condition' and 'smoking.' If these events were independent, smoking would have no effect on the probability of the condition.

Analyze Given Probabilities

We know that 57% of smokers develop the condition and 13% of non-smokers do. Since these percentages differ, the condition is more likely in smokers, indicating dependence between smoking and lung condition.

Calculate Overall Probability of Lung Condition

Utilize the law of total probability. Define S as the event of smoking and C as the condition. We have: - The probability of smoking: \( P(S) = 0.23 \)- The probability of not smoking: \( P(S^c) = 0.77 \)- The probability of having the condition given smoking: \( P(C|S) = 0.57 \)- The probability of having the condition given not smoking: \( P(C|S^c) = 0.13 \)The overall probability of having the condition is calculated as:\[ P(C) = P(C|S)P(S) + P(C|S^c)P(S^c) = (0.57)(0.23) + (0.13)(0.77) \]

Calculate and Interpret

Calculating the above expression, we get:\[ P(C) = (0.57)(0.23) + (0.13)(0.77) = 0.1311 + 0.1001 = 0.2312 \]Hence, the probability that a randomly selected 60-year-old has this lung condition is 23.12%.

Key concepts

Independence in Statistics

In statistics, understanding the concept of independence between two events is crucial. An event A and an event B are considered independent if the occurrence of A does not affect the occurrence of B. This means that the probability of event A occurring, given that event B has occurred, is the same as the probability of event A occurring independently, mathematically represented as \( P(A|B) = P(A) \).

In the context of the provided exercise, we are considering the events 'having a lung condition' (event C) and 'smoking' (event S). If these two events were independent, the probability of developing the lung condition would be the same whether a person smokes or not. However, the given statistics tell a different story, indicating that smokers are more likely (57%) to develop the condition compared to non-smokers (13%). Therefore, these percentages clearly illustrate dependency since the probability of developing the condition is significantly higher for smokers compared to non-smokers.

Law of Total Probability

The Law of Total Probability is a fundamental rule that allows us to evaluate probabilities by considering all possible ways an event can occur. It's especially useful when dealing with complex scenarios where an event can be broken down into several different conditions or subsets.

In this exercise, to determine the overall probability that a 60-year-old has the lung condition, we leverage this law. We need to consider both whether the person smokes and whether they do not smoke, as these are the two mutually exclusive conditions under which the event of developing the lung condition could occur. The law is applied as follows:

• Let \( P(S) \) be the probability of a person smoking, which is 0.23.
• Let \( P(S^c) \) be the probability of not smoking, which is 0.77 (since it is complementary to smoking).
• The probability of having the condition given smoking \( P(C|S) \) is 0.57.
• The probability of having the condition given not smoking \( P(C|S^c) \) is 0.13.

By considering these probabilities, the overall probability \( P(C) \) is computed through the equation:\[P(C) = P(C|S) \times P(S) + P(C|S^c) \times P(S^c)\]This accounts for all individuals, whether they smoke or not, in the calculation for the probability of having the condition.

Probability Calculations

To solve problem b in the exercise, we calculate the overall probability that a randomly selected 60-year-old has the lung condition using the formula derived from the Law of Total Probability. It's crucial to carefully examine each component of the calculation to ensure we are combining the probabilities correctly.

We break down the calculation as follows:

• The probability that a smoker develops the condition is given by \( P(C|S) = 0.57 \).
• The probability that a non-smoker develops the condition is given by \( P(C|S^c) = 0.13 \).
• The probability of being a smoker is \( P(S) = 0.23 \), while the probability of being a non-smoker is \( P(S^c) = 0.77 \).

By applying these to the formula:\[P(C) = (0.57) \times (0.23) + (0.13) \times (0.77)\]This calculation gives us:\[P(C) = 0.1311 + 0.1001 = 0.2312\]Thus, the probability that a randomly selected 60-year-old has the lung condition is 23.12%. This outcome emphasizes how combining individual probabilities can provide a comprehensive understanding of overall likelihood in a population.