https://file.aiquickdraw.com/mathbot/file/1725068186278b8it1me2.jpg

s =2, h = 2, b = 5, sb = 3, bh = 2,  sh = 2, shb = 1

The problem involves finding the probability that a student chosen at random plays baseball and soccer but not hockey. The Venn diagram illustrates the relationships between students who play baseball (B), soccer (S), and hockey (H). The numbers in each section of the Venn diagram correspond to the number of students who participate in those sports:

- \( S \) (only soccer): 2
- \( H \) (only hockey): 2
- \( B \) (only baseball): 5
- \( S \cap B \) (soccer and baseball, but not hockey): 3
- \( B \cap H \) (baseball and hockey, but not soccer): 2
- \( S \cap H \) (soccer and hockey, but not baseball): 2
- \( S \cap H \cap B \) (all three sports): 1

### Steps to Solve:

1. **Determine Total Students:**
   Sum up all the values from the Venn diagram to find the total number of students surveyed:

   \[
   \text{Total} = 2 + 2 + 5 + 3 + 2 + 2 + 1 = 17
   \]

2. **Identify the Target Group:**
   The target group is students who play both baseball and soccer but **not** hockey, which is the section labeled \( S \cap B \) (3 students).

3. **Calculate the Probability:**
   The probability \( P \) is calculated as the number of students in the target group divided by the total number of students:

   \[
   P(\text{B and S, not H}) = \frac{\text{Number of B and S, not H}}{\text{Total number of students}} = \frac{3}{17}
   \]

### Final Answer:
The probability that a randomly chosen student plays baseball and soccer but not hockey is \(\frac{3}{17}\).

---

Would you like more details on any step, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the probability that a randomly chosen student plays only baseball?
2. What is the probability that a student plays both soccer and hockey but not baseball?
3. How would the probability change if the number of students playing all three sports increased by one?
4. What is the probability that a randomly chosen student plays at least one of the sports?
5. How many students play exactly two sports?

**Tip:** Venn diagrams are a powerful tool for visualizing probabilities and set relationships in overlapping groups.

Explore how to calculate the probability that a student plays baseball and soccer but not hockey using Venn diagrams and set theory. Includes step-by-step solution and detailed explanation. Suitable for high school students studying probability and set theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation