A box contains 4 chocolates and 1 fruit chew. Clark and Chloe take turns drawing a treat out of the box without replacement. Whoever draws the fruit chew wins. Clark draws first. What is the probability that Chloe wins? Express your answer as a common fraction.

To solve this problem, we need to calculate the probability that Chloe wins, i.e., the probability that Chloe draws the fruit chew. Since Chloe draws after Clark, we will examine the sequence of draws.

### Step 1: Total number of treats
The box contains 4 chocolates and 1 fruit chew, making a total of 5 treats.

### Step 2: Sequence of turns
- Clark draws first.
- Chloe draws second, and the drawing alternates between Clark and Chloe until the fruit chew is drawn.

### Step 3: Probabilities of Chloe drawing the fruit chew
Chloe wins if she draws the fruit chew. Let's break down the possible scenarios where Chloe can win.

1. **Scenario 1: Chloe draws the fruit chew on her first turn (the second draw overall):**
   - Clark must draw a chocolate on his first turn (probability: \( \frac{4}{5} \), since 4 chocolates are in the box and 1 fruit chew).
   - Then Chloe must draw the fruit chew on her turn (probability: \( \frac{1}{4} \), since 1 fruit chew remains among the 4 remaining treats).
   
   Thus, the probability of this scenario is:
   \[
   \frac{4}{5} \times \frac{1}{4} = \frac{4}{20} = \frac{1}{5}.
   \]

2. **Scenario 2: Chloe draws the fruit chew on her second turn (the fourth draw overall):**
   - Clark and Chloe both need to draw chocolates on the first three turns. The probability of this happening is:
     - Clark draws a chocolate on his first turn: \( \frac{4}{5} \),
     - Chloe draws a chocolate on her first turn: \( \frac{3}{4} \) (since 3 chocolates remain among the 4 treats),
     - Clark draws another chocolate on his second turn: \( \frac{2}{3} \) (since 2 chocolates remain among the 3 treats).
   - Then Chloe must draw the fruit chew on her second turn (probability: \( \frac{1}{2} \), since 1 fruit chew remains among the 2 remaining treats).

   Thus, the probability of this scenario is:
   \[
   \frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} = \frac{24}{120} = \frac{1}{5}.
   \]

### Step 4: Total probability that Chloe wins
The total probability that Chloe wins is the sum of the probabilities from the two scenarios:
\[
\frac{1}{5} + \frac{1}{5} = \frac{2}{5}.
\]

Thus, the probability that Chloe wins is \( \boxed{\frac{2}{5}} \).

Would you like more details on any part of the solution, or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the probability that Clark wins?
2. How does the probability change if there are more chocolates and only one fruit chew?
3. What if Clark and Chloe take more than one treat at a time—how would that affect the outcome?
4. How does the solution change if the treats are replaced after being drawn?
5. Can you generalize this problem to \( n \) chocolates and 1 fruit chew?

**Tip:** Always start by considering all possible outcomes and organizing the sequence of events clearly, especially in multi-stage probability problems.

Explore how to calculate the probability that Chloe wins in a game where Clark and Chloe draw treats from a box without replacement. With a step-by-step breakdown of the process, learn key probability concepts and methods. Ideal for Grades 8-10.

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