2. A bag contains 18 white balls and 9 black balls. If 2 balls are drawn consecutively without replacement. a. Draw the tree diagram for the experiment. b. List all the possible outcomes. c. Find the probability that the two drawn are; i. both white ii. both black

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Let's address each part of the problem step by step.

### **Problem Statement Recap**
- **Total Balls in Bag:** 18 white (W) + 9 black (B) = **27 balls**
- **Experiment:** Drawing **2 balls consecutively without replacement**

---

### **a. Draw the Tree Diagram for the Experiment**

A **tree diagram** visually represents all possible outcomes of a probabilistic experiment. For this scenario, the tree diagram has two levels: one for the first draw and another for the second draw based on the first outcome.

Since we can't draw here, I'll describe it textually:

1. **First Draw:**
   - **Outcome 1:** White (W) with probability $$ \frac{18}{27} $$
   - **Outcome 2:** Black (B) with probability $$ \frac{9}{27} $$

2. **Second Draw:** (Depends on the first outcome)
   - **If First Draw was White (W):**
     - **Second Outcome 1:** White (W) with probability $$ \frac{17}{26} $$ *(since one white ball is already drawn)*
     - **Second Outcome 2:** Black (B) with probability $$ \frac{9}{26} $$
     
   - **If First Draw was Black (B):**
     - **Second Outcome 1:** White (W) with probability $$ \frac{18}{26} $$
     - **Second Outcome 2:** Black (B) with probability $$ \frac{8}{26} $$ *(since one black ball is already drawn)*

**Textual Representation of the Tree Diagram:**

```
First Draw:
├── White (W) [18/27]
│   ├── White (W) [17/26]
│   └── Black (B) [9/26]
└── Black (B) [9/27]
    ├── White (W) [18/26]
    └── Black (B) [8/26]
```

---

### **b. List All the Possible Outcomes**

When drawing two balls consecutively without replacement, the outcomes are **ordered pairs** representing the sequence of colors drawn.

The possible outcomes are:

1. **WW**: First ball White, Second ball White
2. **WB**: First ball White, Second ball Black
3. **BW**: First ball Black, Second ball White
4. **BB**: First ball Black, Second ball Black

So, the set of all possible outcomes is:

$$
\{ \text{WW}, \text{WB}, \text{BW}, \text{BB} \}
$$

---

### **c. Find the Probability that the Two Drawn are:**

#### **i. Both White (WW)**

1. **First Draw:** Probability of drawing a white ball:
   $$
   P(W_1) = \frac{18}{27} = \frac{2}{3}
   $$

2. **Second Draw:** After drawing one white ball, there are now 17 white balls left out of 26 total balls:
   $$
   P(W_2 | W_1) = \frac{17}{26}
   $$

3. **Combined Probability:**
   $$
   P(\text{WW}) = P(W_1) \times P(W_2 | W_1) = \frac{18}{27} \times \frac{17}{26} = \frac{2}{3} \times \frac{17}{26} = \frac{34}{78} = \frac{17}{39} \approx 0.4359 \text{ or } 43.59\%
   $$

#### **ii. Both Black (BB)**

1. **First Draw:** Probability of drawing a black ball:
   $$
   P(B_1) = \frac{9}{27} = \frac{1}{3}
   $$

2. **Second Draw:** After drawing one black ball, there are now 8 black balls left out of 26 total balls:
   $$
   P(B_2 | B_1) = \frac{8}{26} = \frac{4}{13}
   $$

3. **Combined Probability:**
   $$
   P(\text{BB}) = P(B_1) \times P(B_2 | B_1) = \frac{9}{27} \times \frac{8}{26} = \frac{1}{3} \times \frac{4}{13} = \frac{4}{39} \approx 0.1026 \text{ or } 10.26\%
   $$

---

### **Summary of Probabilities**

- **Both White (WW):** $$ \frac{17}{39} $$ ≈ **43.59%**
- **Both Black (BB):** $$ \frac{4}{39} $$ ≈ **10.26%**

---

These calculations provide a clear understanding of the probabilities involved in drawing two balls consecutively without replacement from a bag containing white and black balls.
**a. Tree Diagram:** - First Draw: White (W) or Black (B) - Second Draw: Depends on first draw **b. Possible Outcomes:** 1. WW 2. WB 3. BW 4. BB **c. Probabilities:** - Both White (WW): 43.59% - Both Black (BB): 10.26%

2. A bag contains 18 white balls and 9 black balls. If 2 balls are drawn consecutively without replacement. a. Draw the tree diagram for the experiment. b. List all the possible outcomes. c. Find the probability that the two drawn are; i. both white ii. both black

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