Drama 38 137 51 8 Band Athletics 10 1. How many students are in the senior class? 2. How many students participated in athletics? 3. If a student is randomly chosen, what is the probability that the student participates in athletics or dramat 4. If a student is randomly chosen, what is the probability that the student participates only in drama and band?
The Correct Answer and Explanation is:
Let’s analyze the Venn diagram to answer the questions accurately.
Given Venn Diagram Values:
- Only Drama = 38
- Only Athletics = 137
- Only Band = 51
- Drama & Athletics only = 4
- Drama & Band only = 10
- Band & Athletics only = 8
- All three (Drama, Band, Athletics) = 30
- None of the three = 10
1. How many students are in the senior class?
Add all categories: 38 \text{ (Drama only)} + 137 \text{ (Athletics only)} + 51 \text{ (Band only)} + 4 \text{ (Drama & Athletics)} + 10 \text{ (Drama & Band)} + 8 \text{ (Band & Athletics)} + 30 \text{ (All three)} + 10 \text{ (None)} = 288
Answer: 288 students
2. How many students participated in athletics?
Athletics includes:
- Only Athletics = 137
- Drama & Athletics = 4
- Band & Athletics = 8
- All three = 30
137+4+8+30=179137 + 4 + 8 + 30 = 179
Answer: 179 students
3. Probability student participates in athletics or drama
Athletics or Drama includes:
- Only Athletics = 137
- Only Drama = 38
- Drama & Athletics = 4
- Drama & Band = 10
- Band & Athletics = 8
- All three = 30
Total = 137+38+4+10+8+30=227137 + 38 + 4 + 10 + 8 + 30 = 227
Probability = 227288\frac{227}{288}
Answer: 227288≈0.788\frac{227}{288} \approx 0.788 or 78.8%
4. Probability student participates only in drama and band
Only students in Drama and Band but not Athletics = 10
Probability = 10288≈0.0347\frac{10}{288} \approx 0.0347 or 3.47%
Explanation
This Venn diagram shows student participation in three activities: Drama, Band, and Athletics. Each circle represents one group, with overlapping areas showing students in multiple activities. To solve these problems, we apply basic set theory and probability.
The first step is identifying how many students are in each category. We sum all the distinct areas: those participating in only one activity, those in two overlapping groups, those in all three, and those in none. This gives the total senior class size, which is 288 students.
For the second question, we find everyone involved in athletics. This includes students who only do athletics, those who do athletics and one or two of the other activities. Adding these gives us 179 students.
The third question asks for the probability that a randomly selected student is in either athletics or drama. This requires adding all students who are involved in at least one of the two. We carefully add those in only drama, only athletics, both, and combinations with the third group, ensuring we don’t double-count shared members. The result is a probability of 78.8%.
Lastly, we find the probability of a student being in both drama and band but not athletics. Only the part of the Venn diagram labeled as Drama & Band (without touching Athletics) is counted. That’s 10 students, so the probability is approximately 3.47%.
These types of problems demonstrate how Venn diagrams simplify the visualization and calculation of overlapping data sets, helping to compute probabilities and totals accurately
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