Suppose 80% of citizens in Singapore are smokers. During a year, 50% of smokers suffer from at least one respiratory disease, but only 10% of non-smokers suffer from at least one respiratory disease. Suppose a citizen in Singapore is drawn randomly. Let S be the event that the selected citizen is a smoker. Let R be the event that the selected citizen is suffering from at least one respiratory disease. Given that a citizen has no respiratory disease in a year, what is the probability that s/he is a smoker?

Question

Suppose 80% of citizens in Singapore are smokers. During a year, 50% of smokers suffer from at least one respiratory disease, but only 10% of non-smokers suffer from at least one respiratory disease. Suppose a citizen in Singapore is drawn randomly. Let S be the event that the selected citizen is a smoker. Let R be the event that the selected citizen is suffering from at least one respiratory disease. Given that a citizen has no respiratory disease in a year, what is the probability that s/he is a smoker?

Are events S and R independent if 30% of citizens in Singapore are smokers?

Answer 1

To find the probability that a citizen is a smoker given that they have no respiratory disease, we can make use of Bayes' theorem.

Let's define the events:
S = the event that the selected citizen is a smoker
R = the event that the selected citizen is suffering from at least one respiratory disease

We want to find P(S | ~R), which is the probability that a citizen is a smoker given that they do not have a respiratory disease. The complement of event R, denoted as ~R, represents the absence of respiratory disease.

According to Bayes' theorem, we have:
P(S | ~R) = (P(~R | S) * P(S)) / P(~R)

Now, let's consider the information provided in the question.

Given that 80% of citizens in Singapore are smokers, the probability of being a smoker, P(S), is 0.8.

We are also informed that during a year, 50% of smokers suffer from at least one respiratory disease. Therefore, the probability of having a respiratory disease, given that a citizen is a smoker, P(R | S), is 0.5.

Now, we need to find the probability of not having a respiratory disease, P(~R), which can be calculated using the complement rule:

P(~R) = 1 - P(R)

We are provided with the information that only 10% of nonsmokers suffer from at least one respiratory disease. Since nonsmokers represent 100% - 80% = 20% of the population, we can calculate:

P(R) = (0.2 * 0.1) + (0.8 * 0.5)

This equation represents the probability that a randomly selected citizen has a respiratory disease: the first term accounts for the probability of a nonsmoker having a respiratory disease (0.2 * 0.1), and the second term accounts for the probability of a smoker having a respiratory disease (0.8 * 0.5).

Therefore, we can find P(~R) = 1 - [(0.2 * 0.1) + (0.8 * 0.5)].

Substituting all the known values into Bayes' theorem:

P(S | ~R) = (P(~R | S) * P(S)) / P(~R)
= (0.5 * 0.8) / [1 - (0.2 * 0.1) + (0.8 * 0.5)]

Simplifying the equation gives the probability that a citizen is a smoker given that they do not have a respiratory disease.

Regarding the second part of the question, events S and R are said to be independent if and only if P(S ∩ R) = P(S) * P(R), where P(S ∩ R) represents the probability of events S and R occurring together.

Given that 30% of citizens in Singapore are smokers, the probability of being a smoker, P(S), is 0.3. We are not provided with the probability of having a respiratory disease for the entire population, so we cannot determine if S and R are independent based on the given information.