The table gives us the following information:

Students with musical instruments and hats: 61

Students with musical instruments and no hats: 89

Students without musical instruments and hats: 139

Students without musical instruments and no hats: 211

P(no instrument and no hat)
P(no instrument ∣ no hat)
P(no hat ∣ no instrument)
Are the events carrying a musical instrument and wearing a hat independent?

Given the corrected dataGiven the corrected 

- Students with musical instruments and hats: 61
- Students with musical instruments and no hats: 89
- Students without musical instruments and hats: 139
- Students without musical instruments and no hats: 211

### Total Number of Students Observed
To find the total number of students observed, we add up all the values in the table:

\[
61 + 89 + 139 + 211 = 500
\]

### Probabilities
Let's calculate the required probabilities:

1. **P(no instrument and no hat)**
   This is the probability that a student is not carrying a musical instrument and not wearing a hat:
   
   \[
   P(\text{no instrument and no hat}) = \frac{\text{Number of students with no instrument and no hat}}{\text{Total number of students}} = \frac{211}{500}
   \]

2. **P(no instrument ∣ no hat)**
   This is the conditional probability that a student is not carrying a musical instrument given that they are not wearing a hat:
   
   \[
   P(\text{no instrument | no hat}) = \frac{\text{Number of students with no instrument and no hat}}{\text{Number of students with no hat}} = \frac{211}{89 + 211} = \frac{211}{300}
   \]

3. **P(no hat ∣ no instrument)**
   This is the conditional probability that a student is not wearing a hat given that they are not carrying a musical instrument:
   
   \[
   P(\text{no hat | no instrument}) = \frac{\text{Number of students with no instrument and no hat}}{\text{Number of students with no instrument}} = \frac{211}{139 + 211} = \frac{211}{350}
   \]

### Independence of Events
To determine if the events of carrying a musical instrument and wearing a hat are independent, we need to check if:

\[
P(\text{instrument and hat}) = P(\text{instrument}) \cdot P(\text{hat})
\]

First, we calculate the individual probabilities:

- \( P(\text{instrument}) = \frac{61 + 89}{500} = \frac{150}{500} = 0.3 \)
- \( P(\text{hat}) = \frac{61 + 139}{500} = \frac{200}{500} = 0.4 \)

Next, we calculate the combined probability assuming independence:

\[
P(\text{instrument}) \cdot P(\text{hat}) = 0.3 \times 0.4 = 0.12
\]

Now compare this to the actual combined probability:

\[
P(\text{instrument and hat}) = \frac{61}{500} = 0.122
\]

If these values are equal, the events are independent. If not, they are not independent.

Since \(0.12 \approx 0.122\), they are very close, but not exactly equal. However, for practical purposes, they are considered to be approximately independent.

### Summary of Results
- Total number of students observed: 500
- \(P(\text{no instrument and no hat}) = \frac{211}{500} = 0.422\)
- \(P(\text{no instrument | no hat}) = \frac{211}{300} \approx 0.703\)
- \(P(\text{no hat | no instrument}) = \frac{211}{350} \approx 0.603\)
- Carrying a musical instrument and wearing a hat are **approximately independent**.

Would you like details or have any questions?

#### Related Questions:
1. What is the probability of a student carrying a musical instrument?
2. What is the probability of a student wearing a hat?
3. What is the probability of a student carrying a musical instrument given that they are wearing a hat?
4. What is the probability of a student not wearing a hat?
5. How many students are carrying musical instruments?
6. How many students are wearing hats?
7. What is the probability of a student wearing a hat given that they are carrying a musical instrument?
8. Are the events of not carrying a musical instrument and not wearing a hat independent?

#### Tip:
When assessing independence, a small difference in calculated probabilities can often be considered negligible in practice, but always clarify the context and significance of this approximation.

Explore the concept of probability and independence of events through a scenario involving students with musical instruments and hats. Calculate probabilities like P(no instrument and no hat), P(no instrument | no hat), and P(no hat | no instrument), and determine if carrying a musical instrument and wearing a hat are independent events.

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