https://file.aiquickdraw.com/mathbot/file/1726915045099pzunt5sh.jpg

We are given that Libby has 3 white counters, 16 black counters, and 1 purple counter, for a total of \(3 + 16 + 1 = 20\) counters. She takes one counter at random from the box, replaces it, and then takes another counter. We are asked to find the probability that Libby takes at least one white counter in two draws.

### Step 1: Probabilities of White and Not White Counters
- The probability of drawing a white counter is \( \frac{3}{20} \).
- The probability of drawing a not white counter (either black or purple) is \( \frac{17}{20} \).

### Step 2: Probability Tree Diagram Outcomes
The tree diagram shows the following outcomes:

1. White on both draws: 
   \[
   P(\text{White, White}) = \frac{3}{20} \times \frac{3}{20} = \frac{9}{400}
   \]

2. White on first, not white on second: 
   \[
   P(\text{White, Not White}) = \frac{3}{20} \times \frac{17}{20} = \frac{51}{400}
   \]

3. Not white on first, white on second: 
   \[
   P(\text{Not White, White}) = \frac{17}{20} \times \frac{3}{20} = \frac{51}{400}
   \]

4. Not white on both draws:
   \[
   P(\text{Not White, Not White}) = \frac{17}{20} \times \frac{17}{20} = \frac{289}{400}
   \]

### Step 3: Calculating the Probability of At Least One White
The complement of "at least one white" is "no white counters," which occurs if both draws are not white. Therefore, we first find the probability of no white counters (not white on both draws):

\[
P(\text{No White}) = \frac{289}{400}
\]

Now, the probability of at least one white counter is the complement of this:

\[
P(\text{At Least One White}) = 1 - P(\text{No White}) = 1 - \frac{289}{400} = \frac{111}{400}
\]

### Final Answer:
The probability that Libby takes at least one white counter is \( \frac{111}{400} \).

---

Would you like further details or clarifications on this? 

Here are 5 related questions to explore this concept further:
1. What is the probability that Libby takes exactly one white counter in two draws?
2. How does the probability change if there are 4 white counters instead of 3?
3. How would the tree diagram change if the counters were not replaced between draws?
4. What is the probability of drawing a purple counter in both draws?
5. Can you extend this problem to three draws and calculate the probability of at least one white counter?

**Tip**: Understanding tree diagrams can help break down complex probability problems into simpler, manageable parts.

Libby has 3 white counters, 16 black counters, and 1 purple counter in a box. She takes one counter at random, replaces it, and takes another. Find the probability that Libby takes at least one white counter.

Learn how to calculate the probability of drawing at least one white counter when Libby picks counters from a box containing 3 white, 16 black, and 1 purple counters. Using a tree diagram and complementary events, this problem explores basic probability concepts. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation