A campus radio station surveyed 500 students to determine the types of music they like.  The survey revealed that 207 like rock, 163 like country, and 119 like jazz.  Moreover, 31 like rock and country, 27 like rock and jazz, 23 like country and jazz, and 11 like all three types of music.  What is the probability that a randomly selected student likes at least two of the three types of music?
 0.4000


 0.0940


 0.0220


 0.3020


 0.1180


 None of the above.


To find the probability that a randomly selected student likes at least two of the three types of music, we’ll use the principle of inclusion-exclusion.

Let:
- \( R \) be the set of students who like rock,
- \( C \) be the set of students who like country,
- \( J \) be the set of students who like jazz.

Given values are:
- \( |R| = 207 \)
- \( |C| = 163 \)
- \( |J| = 119 \)
- \( |R \cap C| = 31 \)
- \( |R \cap J| = 27 \)
- \( |C \cap J| = 23 \)
- \( |R \cap C \cap J| = 11 \)

We want to find the number of students who like **at least two types** of music.

### Step 1: Calculate students who like at least two types of music
To get students who like at least two types, we add the numbers who like exactly two types and those who like all three:

1. **Students who like exactly two types of music**:
   - Rock and Country only: \( |R \cap C| - |R \cap C \cap J| = 31 - 11 = 20 \)
   - Rock and Jazz only: \( |R \cap J| - |R \cap C \cap J| = 27 - 11 = 16 \)
   - Country and Jazz only: \( |C \cap J| - |R \cap C \cap J| = 23 - 11 = 12 \)

2. **Students who like all three types of music**:
   - \( |R \cap C \cap J| = 11 \)

3. **Total students who like at least two types**:
   \[
   20 + 16 + 12 + 11 = 59
   \]

### Step 2: Calculate the probability
The probability that a randomly selected student likes at least two types of music is:
\[
\frac{\text{Number of students who like at least two types}}{\text{Total number of students}} = \frac{59}{500} = 0.118
\]

### Conclusion
The probability is \( \mathbf{0.1180} \).

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Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How would the probability change if more students liked only one type of music?
2. How does the inclusion-exclusion principle apply to other overlapping sets in probability?
3. What if the total student count changes but preferences remain the same?
4. Can we calculate the probability of students liking exactly one type of music?
5. How would this probability change if all students liked at least one type of music?

**Tip:** When using the inclusion-exclusion principle, ensure that all possible overlapping sets are accounted for to avoid over-counting or under-counting any groups.

A campus radio station surveyed 500 students to determine the types of music they like. The survey revealed that 207 like rock, 163 like country, and 119 like jazz. Moreover, 31 like rock and country, 27 like rock and jazz, 23 like country and jazz, and 11 like all three types of music. What is the probability that a randomly selected student likes at least two of the three types of music?
0.4000
0.0940
0.0220
0.3020
0.1180
None of the above.

Learn how to calculate the probability that a randomly selected student from a survey of 500 likes at least two types of music. This problem involves the inclusion-exclusion principle and probability concepts, providing a step-by-step breakdown of overlapping sets in music preferences.

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