Question

Suppose that in a certain population of married couples, \(30 \%\) of the husbands smoke, \(20 \%\) of the wives smoke, and in \(8 \%\) of the couples both the husband and the wife smoke. Is the smoking status (smoker or nonsmoker) of the husband independent of that of the wife? Why or why not?

Short answer

No, the smoking status of the husband is not independent of that of the wife, because the joint probability of both smoking is not equal to the product of the individual probabilities (0.08 != 0.06).

Step-by-step solution

Understand the concept of independence

Two events A and B are independent if the probability of A occurring is the same whether or not B occurs. Mathematically, A and B are independent if and only if the probability of both A and B occurring is equal to the product of their individual probabilities, which means P(A and B) = P(A) * P(B).

Identify the given probabilities

The problem provides the probability that the husband smokes, denoted as P(H) = 0.30, the probability that the wife smokes, denoted as P(W) = 0.20, and the probability that both the husband and wife smoke, denoted as P(H and W) = 0.08.

Apply the independence criterion

To check if the smoking status is independent between husband and wife, calculate the product of the individual probabilities P(H) * P(W) and compare it with the joint probability P(H and W).

Calculate the product of individual probabilities

Compute the product of the probability that the husband smokes and the probability that the wife smokes: P(H) * P(W) = 0.30 * 0.20 = 0.06.

Compare the calculated product with the joint probability

Compare the product of individual probabilities P(H) * P(W) = 0.06 with the given joint probability P(H and W) = 0.08.

Conclude the independence

Since P(H and W) = 0.08 is not equal to P(H) * P(W) = 0.06, the smoking status of the husband is not independent of that of the wife. The events are dependent because knowing the smoking status of one influences the probability of the smoking status of the other.

Key concepts

Event Independence

Understanding event independence is essential for various probability calculations. Two events are considered independent when the outcome of one event does not affect the outcome of another. Formally, we say events A and B are independent if the probability of A occurring does not change whether B occurs or not; this is expressed mathematically as:
\[ P(A \text{ and } B) = P(A) \times P(B) \].
In the provided exercise, we looked to determine if the smoking status of husbands was independent of that of their wives in a population. Independence would mean that a husband's likelihood of smoking would not be influenced by whether his wife smokes, and vice versa. To verify this, we compared the joint probability of both smoking against the product of their individual smoking probabilities.

Joint Probability

The joint probability is a statistical measure that calculates the likelihood of two events happening at the same time. It's denoted as
\( P(A \text{ and } B) \),
and is fundamental in understanding the relationship between events. For independent events, the joint probability should equal the product of the probabilities of each event occurring separately. In our example, we calculated the joint probability of both the husband and the wife smoking, which was provided as
\( P(H \text{ and } W) = 0.08 \).
Comparing this to the product of their individual probabilities was key in determining their independence.

Probability Calculation

Probability calculation involves quantifying the chances of an event occurring. For a single event, it is simply the ratio of favorable outcomes to the total possible outcomes. When it comes to multiple events, calculations can become complex, involving additional rules and considerations, particularly around independence and joint probabilities.
In the scenario with smoking habits, we first calculated the product of the individual probabilities for each of the spouses smoking:
\( P(H) \times P(W) = 0.30 \times 0.20 \),
resulting in
\(0.06\).
Since this product did not match the joint probability of both spouses smoking, we concluded that these events are dependent. It's essential for students to grasp this process as it demonstrates a practical application of probability calculations in evaluating the dependency between events.