https://file.aiquickdraw.com/mathbot/file/1728307589442odz3duhx.jpg

Let's solve the probability questions based on the provided table.

### Information from the table:
- There are **288 marbles** in total.
- The marbles are categorized based on **color** (Red or Green) and **material** (Glass, Steel, or Plastic).

|           | Glass | Steel | Plastic | Total |
|-----------|-------|-------|---------|-------|
| **Red**   | 20    | 13    | 25      | 45    |
| **Green** | 108   | 60    | 135     | 243   |
| **Total** | 128   | 73    | 160     | 288   |

Now, let's go question by question:

---

### a) What is the probability that the marble is Red and made of Glass?

From the table, there are **20** Red marbles made of Glass. The total number of marbles is **288**.

So, the probability is:
\[
P(\text{Red and Glass}) = \frac{20}{288} = \frac{5}{72}
\]

---

### b) What is the probability that the marble is Red?

There are **45** Red marbles in total (sum of Red marbles across all materials). The total number of marbles is **288**.

So, the probability is:
\[
P(\text{Red}) = \frac{45}{288} = \frac{15}{96} = \frac{5}{32}
\]

---

### c) What is the probability that the marble is made of Glass?

There are **128** marbles made of Glass in total. The total number of marbles is **288**.

So, the probability is:
\[
P(\text{Glass}) = \frac{128}{288} = \frac{32}{72} = \frac{8}{18} = \frac{4}{9}
\]

---

### d) What is the probability that the marble is Red given that it is made of Glass?

This is a conditional probability. We want \( P(\text{Red} | \text{Glass}) \), the probability that the marble is Red given that it is made of Glass. The formula for conditional probability is:

\[
P(\text{Red} | \text{Glass}) = \frac{P(\text{Red and Glass})}{P(\text{Glass})}
\]

From earlier:
- \( P(\text{Red and Glass}) = \frac{5}{72} \)
- \( P(\text{Glass}) = \frac{4}{9} \)

So:
\[
P(\text{Red} | \text{Glass}) = \frac{\frac{5}{72}}{\frac{4}{9}} = \frac{5}{72} \times \frac{9}{4} = \frac{45}{288} = \frac{5}{32}
\]

---

### e) What is the probability that the marble is made of Glass given that it is Red?

This is also a conditional probability. We want \( P(\text{Glass} | \text{Red}) \), the probability that the marble is made of Glass given that it is Red. Again, we use the formula for conditional probability:

\[
P(\text{Glass} | \text{Red}) = \frac{P(\text{Red and Glass})}{P(\text{Red})}
\]

From earlier:
- \( P(\text{Red and Glass}) = \frac{5}{72} \)
- \( P(\text{Red}) = \frac{5}{32} \)

So:
\[
P(\text{Glass} | \text{Red}) = \frac{\frac{5}{72}}{\frac{5}{32}} = \frac{5}{72} \times \frac{32}{5} = \frac{160}{288} = \frac{5}{9}
\]

---

### Summary of Answers:
a) \( \frac{5}{72} \)  
b) \( \frac{5}{32} \)  
c) \( \frac{4}{9} \)  
d) \( \frac{5}{32} \)  
e) \( \frac{5}{9} \)

Let me know if you'd like more details on any part of the solution!

---

**Follow-up questions:**
1. How would you calculate the probability if the marble was Green and made of Steel?
2. What is the probability that a randomly chosen marble is neither Red nor made of Steel?
3. What would change in the calculations if we added more categories (e.g., Blue marbles)?
4. How does conditional probability differ from basic probability?
5. What is the complement of the probability of choosing a Red marble?

**Tip:** Conditional probability helps us narrow down the outcomes by focusing on a specific condition, which can be crucial in multi-category problems like this one.

Suppose a marble is drawn at random from a jar containing 288 marbles. The marbles are either Red or Green, and made from either Glass, Steel, or Plastic. What is the probability that: a) The marble is Red and made of Glass? b) The marble is Red? c) The marble is made of Glass? d) The marble is Red given that it is made of Glass? e) The marble is made of Glass given that it is Red?

Learn how to calculate the probability of drawing a Red and Glass marble from a jar of 288 marbles. This example covers both basic and conditional probability calculations. The marble categories are based on color (Red or Green) and material (Glass, Steel, Plastic).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation