Question

The Australian newspaper The Mercury (May 30,1995\()\) reported that based on a survey of 600 reformed and current smokers, \(11.3 \%\) of those who had attempted to quit smoking in the previous 2 years had used a nicotine aid (such as a nicotine patch). It also reported that \(62 \%\) of those who had quit smoking without a nicotine aid began smoking again within 2 weeks and \(60 \%\) of those who had used a nicotine aid began smoking again within 2 weeks. If a smoker who is trying to quit smoking is selected at random, are the events selected smoker who is trying to quit uses a nicotine aid and selected smoker who has attempted to quit begins smoking again within 2 weeks independent or dependent events? Justify your answer using the given information.

Short answer

The events that a selected smoker who is trying to quit uses a nicotine aid and begins smoking again within two weeks are dependent events.

Step-by-step solution

Define the events

Let us define the following events: Event A is a smoker uses a nicotine aid. Event B is a smoker starts smoking again within two weeks. From the problem, it's given that the probability of each event occurring is as follows: \(P(A) = 0.113\), \(P(B) = 0.62\) without a nicotine aid and \(0.60\) with a nicotine aid.

Determine the joint probability

Since only those who quit smoking are considered for restarting smoking again, the chances of a smoker using a nicotine aid and starting smoking again is given by the smokers who used a nicotine aid and started smoking which is \(0.113 \times 0.60 = 0.0678\), denoted \(P(A \cap B)\).

Check for independence

Two events A and B are independent if and only if \(P(A \cap B) = P(A) \times P(B)\). We know \(P(A \cap B) = 0.0678\), and \(P(A) \times P(B) = 0.113 \times 0.62 = 0.07006\). Since these two quantities are not equal, the two events A and B are dependent.