Independent?
Refer to the contingency table in Problem 17 that relates age and likelihood to buy American. Determine P(45–54 years old) and P(45–54 years old | more likely). Are the events "45–54 years old" and "more likely" independent?

Verified step by step guidance

1. Identify the total number of observations in the contingency table by summing all the frequencies across all age groups and likelihood categories. This total will be used to calculate probabilities.
2. Calculate the probability of being 45–54 years old, denoted as \(P(45\text{–}54)\), by summing the frequencies of all individuals aged 45–54 across all likelihood categories and dividing by the total number of observations.
3. Calculate the conditional probability \(P(45\text{–}54 \mid \text{more likely})\) by taking the number of individuals aged 45–54 who are 'more likely' to buy American and dividing by the total number of individuals who are 'more likely' to buy American.
4. Recall that two events A and B are independent if and only if \(P(A \mid B) = P(A)\). In this context, check if \(P(45\text{–}54 \mid \text{more likely})\) equals \(P(45\text{–}54)\).
5. Compare the two probabilities calculated in steps 2 and 3. If they are equal, conclude that the events '45–54 years old' and 'more likely' are independent; if not, they are dependent.

Key Concepts

Probability of an Event

The probability of an event is the measure of the likelihood that the event will occur, calculated as the ratio of favorable outcomes to the total number of outcomes. For example, P(45–54 years old) represents the proportion of individuals aged 45–54 in the entire sample.

Conditional Probability

Conditional probability, denoted P(A|B), is the probability of event A occurring given that event B has occurred. It is found by dividing the probability of both events happening together by the probability of event B. Here, P(45–54 years old | more likely) measures the chance of being 45–54 years old among those more likely to buy American.

Independence of Events

Two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if P(A|B) = P(A). To determine independence here, compare P(45–54 years old) with P(45–54 years old | more likely).