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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.

Step by step solution

01

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.

02

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 \(P(A \cap B)\).

03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events

Independent events are an essential concept within probability that describes scenarios where the occurrence of one event does not affect the likelihood of another event happening. In simpler terms, two events are independent if they have no influence over each other.

For example, consider flipping a coin and rolling a die. The result of the coin flip does not affect the outcome of the die roll.

• This means the probability of getting heads on the coin (let's say 0.5) has no impact on the probability of rolling a 4 on the die (1/6).
• If the events are independent, the joint probability, or the probability that both events will occur, is the product of their individual probabilities.

Mathematically, two events A and B are independent if: \[ P(A \cap B) = P(A) \times P(B) \]In our exercise, the events do not satisfy this condition due to differing joint probability values, indicating they are not independent.

Dependent Events

In probability, dependent events are those where the occurrence of one event does affect the probability of another. The outcome of one event impacts the likelihood of another event happening.

For example, when you draw a card from a deck and do not replace it before drawing another, the result of the first draw alters the probabilities for the second draw.

• If you draw an Ace and don't replace it, the probability of drawing another Ace on the second draw changes because fewer Aces remain in the deck.
• This results in a joint probability different from what would be if both events were independent.

Returning to our exercise, we found that the probability of both using a nicotine aid and starting smoking again (joint probability) did not align with the product of their individual probabilities. Hence, they must be dependent events.

Joint Probability

Joint probability is a fundamental concept in probability theory that refers to the probability of two events happening at the same time. It answers questions like, "What are the chances of both Event A and Event B occurring?"

Calculating joint probability involves multiplying the probabilities of two independent events. However, when events are dependent, it requires more information about how the events relate to each other.

• For independent events: \( P(A \cap B) = P(A) \times P(B) \)
• For dependent events, the joint probability cannot simply be derived as the product of individual probabilities without considering their dependency.

In the exercise, we calculate the joint probability of a smoker using a nicotine aid and relapsing. The outcome reveals dependency, since the calculation showed a disparity from the expected result if they were independent.