- Identify the total count of female students (\(50\)), liberal arts majors (\(38\)), and the overlap (\(18\)).
- Apply the inclusion-exclusion principle: \(n(\text{Female} \cup \text{Liberal Arts}) = 50 + 38 - 18 = 70\).
- Calculate the probability: \(P = \frac{70}{100} = \frac{7}{10}\).
- The final answer is \(\boxed{\frac{7}{10}}\).
Explanation
- Analyze the data provided in the table
We are analyzing a survey of \(100\) students regarding their gender and major. To determine the probability that a randomly selected student is either a female or a liberal arts major, we use the principle of inclusion-exclusion. The total number of students is \(N = 100\). Based on the provided table:
- The number of female students is \(n(\text{Female}) = 15 + 18 + 17 = 50\).
- The number of students majoring in Liberal Arts is \(n(\text{Liberal Arts}) = 20 + 18 = 38\).
- The number of students who are both female and majoring in Liberal Arts is \(n(\text{Female} \cap \text{Liberal Arts}) = 18\).
- Apply the principle of inclusion-exclusion
To find the total number of students who meet at least one of the criteria (being female or being a liberal arts major), we use the addition rule for probabilities, or the inclusion-exclusion principle for counts:
\[n(\text{Female} \cup \text{Liberal Arts}) = n(\text{Female}) + n(\text{Liberal Arts}) - n(\text{Female} \cap \text{Liberal Arts})\]
Substituting the values we identified:
\[n(\text{Female} \cup \text{Liberal Arts}) = 50 + 38 - 18 = 70\]
So, there are \(70\) students who are either female or majoring in liberal arts.
- Calculate the probability and simplify the result
The probability \(P\) of picking a student who is either a female or a liberal arts major is the ratio of the number of favorable outcomes to the total number of students:
\[P(\text{Female} \cup \text{Liberal Arts}) = \frac{n(\text{Female} \cup \text{Liberal Arts})}{N} = \frac{70}{100}\]
Simplifying the fraction \(\frac{70}{100}\) by dividing both the numerator and denominator by their greatest common divisor, which is \(10\), we get:
\[\frac{70}{100} = \frac{7}{10}\]
Comparing this result to the given options, we find it matches option (b).
Examples
Understanding basic probability and set operations is essential for analyzing survey data, such as determining market demographics or student populations. For instance, if you want to know the proportion of customers who belong to two overlapping target segments—like "Age group A" and "Purchased product X"—you use the same inclusion-exclusion method to avoid double-counting. For example, if you have 100 customers, 40 are in "Age group A," 30 bought "Product X," and 10 are both, the number of unique customers reached is \(40 + 30 - 10 = 60\), leading to a probability of \(\frac{60}{100} = 0.6\).