Question

A survey of people in a given region showed that \(20 \%\) were smokers. The probability of death due to lung cancer, given that a person smoked, was roughly 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is .006, what is the probability of death due to lung cancer given that a person is a smoker?

Short answer

Answer: The probability of a person dying due to lung cancer given that they are a smoker is approximately 0.8571 or \(\frac{6}{7}\).

Step-by-step solution

Define notations and problem statement

Let S be the event that a person is a smoker, and L be the event that a person dies of lung cancer. We are given P(S) = 0.20, and the probability of death due to lung cancer for the entire region is P(L) = 0.006. We are asked to find the probability P(L | S) - the probability of death due to lung cancer given that a person is a smoker.

Use Bayes' theorem to write the expression for P(L | S)

Bayes' theorem states that P(L | S) = P(S | L)*P(L)/P(S). However, we don't have the values for P(S | L) and P(S). We will determine these values using the given information.

Calculate P(S' | L) and P(S' | L')

Let S' be the event of a person being a non-smoker, and L' be the event of a person not dying of lung cancer. We are given that the probability of death due to lung cancer for a smoker is 10 times that of a non-smoker. So, P(S | L) = 10P(S' | L). We know that P(S) + P(S') = 1. Thus, P(S') = 1 - P(S) = 1 - 0.20 = 0.80.

Calculate P(L | S')

We know that P(L) = P(L | S)P(S) + P(L | S')P(S'). We can re-write this equation to find P(L | S'): P(L | S') = (P(L) - P(L | S)P(S)) / P(S').

Substitute the values and find P(L | S)

Now, we can use the values obtained in step 3 and 4 to express P(L | S) in terms of known values: $$ P(L | S) = \frac{10P(S' | L)*P(L)}{P(S)} = \frac{10((P(L) - P(L | S)P(S)) / P(S'))*P(L)}{P(S)}. $$ Plugging in the given values: $$ P(L | S) = \frac{10((0.006 - 0.007P(L | S))*0.006)}{0.20} = \frac{0.036 - 0.042P(L | S)}{0.20}. $$ Now, we can solve for P(L | S): $$ 0.180 = 0.36 - 0.21P(L | S) $$ $$ P(L | S) = \frac{0.36 - 0.18}{0.21} = \frac{0.18}{0.21} = \frac{6}{7}, $$ Thus, the probability of a person dying due to lung cancer given that they are a smoker P(L | S) is \(\frac{6}{7}\) or approximately 0.8571.

Key concepts

Bayes' theorem

Bayes' theorem is a powerful mathematical formula used for calculating the conditional probability of an event. The theorem is named after the Reverend Thomas Bayes and provides a way to revise existing predictions or theories (update probabilities) given new or additional evidence.

In the context of our exercise, Bayes' theorem allows us to compute the probability that someone will die due to lung cancer given they are a smoker, using the information already known about the general population and smokers. Bayes' theorem is expressed in the formula:
\[\begin{equation}P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}d{equation}
Here, \( P(A | B) \) is the conditional probability of event A occurring given that event B is true, \( P(B | A) \) is the conditional probability of event B given event A, \( P(A) \) is the probability of event A, and \( P(B) \) is the probability of event B.

Conditional probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. This concept is crucial for understanding complex probability questions where events are interdependent.

In our exercise, we're focusing on the conditional probability that a person dies of lung cancer given they're a smoker. Notationally, this is represented as \( P(L | S) \), where L represents the event of dying from lung cancer and S represents the event of being a smoker.

To find this, we're considering how the probability of lung cancer changes under the condition that the person is a smoker. It's important to recognize that conditional probability helps us isolate the impact of one specific factor (smoking) on the probability of an outcome (lung cancer) by controlling for that factor in our calculations.

Probability notations

Probability notations are the symbols and abbreviations used to represent probabilities, events, and related concepts in mathematical expressions. Understanding these notations is essential for effectively communicating and solving probability problems.

In probability notation:

• \( P(A) \) represents the probability of event A
• \( P(A | B) \) is the conditional probability of A given B
• \( P(A \'cap' B) \) signifies the probability of both A and B occurring
• \( P(A \'cup' B) \) indicates the probability of either A or B (or both) occurring
• \( P(A') \) represents the probability of the complement of A, i.e., A not occurring

These notations were used throughout the solution to express complex relationships between different events—specifically, the likelihood of lung cancer deaths among smokers and non-smokers in the given population.