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.