- Calculate the marginal probability of \(Z\): \(P(Z) = \frac{297}{660} = 0.45\).
- Calculate the conditional probability of \(Z\) given \(B\): \(P(Z|B) = \frac{126}{280} = 0.45\).
- Compare the probabilities to verify the condition for independence: \(P(Z|B) = P(Z)\).
- Conclude that \(Z\) and \(B\) are independent events: \(\boxed{\text{Z and B are independent events because } P(Z|B) = P(Z)}\).
Explanation
- Understand the independence condition
To determine if the events \(Z\) and \(B\) are independent, we use the definition of independence based on probabilities. Two events \(Z\) and \(B\) are considered independent if the occurrence of event \(B\) does not affect the probability of event \(Z\). Mathematically, this is expressed by the condition \(P(Z|B) = P(Z)\), where \(P(Z|B)\) is the conditional probability of \(Z\) given \(B\), and \(P(Z)\) is the marginal probability of \(Z\).
- Calculate the relevant probabilities
From the provided table, we identify the necessary frequencies:
- The total number of observations (the grand total) is \(660\).
- The total number of observations in row \(B\) is \(N(B) = 280\).
- The total number of observations in column \(Z\) is \(N(Z) = 297\).
- The number of observations that are in both row \(B\) and column \(Z\) is \(N(Z \cap B) = 126\).
Using these values, we calculate the marginal probability \(P(Z)\):
\[P(Z) = \frac{N(Z)}{\text{Total}} = \frac{297}{660} = 0.45\]
Next, we calculate the conditional probability \(P(Z|B)\):
\[P(Z|B) = \frac{N(Z \cap B)}{N(B)} = \frac{126}{280} = 0.45\]
- Compare and conclude independence
Comparing the two calculated probabilities:
- \(P(Z) = 0.45\)
- \(P(Z|B) = 0.45\)
Since \(P(Z|B) = P(Z)\), we conclude that the occurrence of event \(B\) does not change the probability of event \(Z\). Therefore, the events \(Z\) and \(B\) are independent.
Examples
In market research, you might use this to determine if a customer's choice of a specific product brand (e.g., Brand B) is independent of their demographic region (e.g., Region Z). If the probability of choosing Brand B remains the same regardless of whether the customer is in Region Z or not, you know the variables are independent, which simplifies your marketing strategy. Similarly, here we checked that \(P(Z|B) = \frac{126}{280} = 0.45\) and \(P(Z) = \frac{297}{660} = 0.45\), proving independence.