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 find the probability that Chloe wins, let's consider the sequence of draws and the specific draw on which the fruit chew is picked. 

### Key Points:
- There are 5 treats in total: 4 chocolates and 1 fruit chew.
- Clark draws first, and then Chloe, alternating turns.
- The game ends when the fruit chew is drawn, so we need to calculate the probability that Chloe is the one who draws it.

### Possible Scenarios:
Chloe wins if she draws the fruit chew on her turn. Let's examine the scenarios where this happens:

1. **Scenario 1:** Clark draws a chocolate on the first turn, then Chloe draws the fruit chew on the second turn.
2. **Scenario 2:** Clark draws a chocolate on the first turn, Chloe draws a chocolate on the second turn, and then Clark draws a chocolate on the third turn, finally Chloe draws the fruit chew on the fourth turn.

These are the only scenarios where Chloe wins because if the game reaches Chloe's turn again after her second draw, all chocolates would already be taken, leaving the fruit chew for Clark on his third draw.

### Calculate the Probability for Each Scenario:

1. **Scenario 1:**
    - Probability that Clark draws a chocolate first: \(\frac{4}{5}\).
    - Probability that Chloe draws the fruit chew next: \(\frac{1}{4}\).

    Probability for Scenario 1:
    \[
    P(\text{Scenario 1}) = \frac{4}{5} \times \frac{1}{4} = \frac{4}{20} = \frac{1}{5}
    \]

2. **Scenario 2:**
    - Probability that Clark draws a chocolate first: \(\frac{4}{5}\).
    - Probability that Chloe draws a chocolate second: \(\frac{3}{4}\).
    - Probability that Clark draws a chocolate third: \(\frac{2}{3}\).
    - Probability that Chloe draws the fruit chew next: \(\frac{1}{2}\).

    Probability for Scenario 2:
    \[
    P(\text{Scenario 2}) = \frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} = \frac{4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2} = \frac{24}{120} = \frac{1}{5}
    \]

### Total Probability that Chloe Wins:
The total probability is the sum of the probabilities of the two scenarios:

\[
P(\text{Chloe wins}) = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}
\]

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

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Would you like further details or have any other questions?

Here are 5 related questions to enhance your understanding:

1. What is the probability that Clark wins on his first draw?
2. If the order of the players were reversed, what would be the probability that Clark wins?
3. What is the probability that the fruit chew is drawn in the third turn?
4. How would the probability change if there were 3 chocolates and 1 fruit chew?
5. If there were 2 fruit chews and 3 chocolates, what would be the probability that Chloe wins?

**Tip:** When dealing with problems involving sequential events, considering each scenario step-by-step and using conditional probabilities can simplify complex probability calculations.

Explore a probability problem involving Clark and Chloe drawing treats from a box with chocolates and a fruit chew. Calculate the probability that Chloe wins, considering sequential draws and conditional probabilities. Suitable for Grades 7-9.

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