- Identify the individual probabilities from the table: \(P(\text{East}) = 0.12\) and \(P(\text{West}) = 0.18\).
- Recognize that living in the East and living in the West are mutually exclusive events.
- Apply the addition rule for mutually exclusive events: \(P(\text{East or West}) = 0.12 + 0.18\).
- Conclude that the total probability is \(\boxed{0.3}\).
Explanation
- Analyze the given data and problem context
To find the probability that a person lives in either the East or the West region, we look at the provided probability distribution table. The table shows that the probability of living in the East region is \(P(\text{East}) = 0.12\) and the probability of living in the West region is \(P(\text{West}) = 0.18\). These events are mutually exclusive because a person is considered to reside in only one of the listed regions.
- Define the mathematical approach
Since the events of living in the East region and the West region are mutually exclusive, we can determine the probability of either event occurring by summing their individual probabilities. The rule for the probability of the union of mutually exclusive events is:
\[P(\text{East or West}) = P(\text{East}) + P(\text{West})\]
- Calculate the final probability
Using the values from the table:
\[P(\text{East or West}) = 0.12 + 0.18\]
Performing the addition, we get:
\[P(\text{East or West}) = 0.30\]
Therefore, the probability that a randomly chosen person lives in either the East or West region is \(0.3\). This corresponds to the fourth option in the provided list.
Examples
This method of summing probabilities for mutually exclusive events is fundamental in statistics. For example, if you are analyzing market research data for a clothing brand, you might want to know the total probability that a customer belongs to one of two distinct age groups to plan an advertising campaign effectively. By identifying that these groups do not overlap, you can simply add their proportions together to find the total target audience share. For instance, if 15% of your customers are in the 18-24 age group and 20% are in the 25-34 group, the probability of a customer being between 18 and 34 is \(\frac{15}{100} + \frac{20}{100} = \frac{35}{100} = 0.35\).