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

It seems like the question got cut off at the end, but based on the information you provided, it appears you're asking for the probability that Clark wins the game, given that he draws first, and they are drawing treats without replacement until someone draws the fruit chew.

Let's break it down step-by-step:

- The box contains 4 chocolates and 1 fruit chew, so there are 5 treats total.
- The goal is to find the probability that **Clark** wins (i.e., Clark draws the fruit chew first).
- Clark draws first, Chloe draws second, and so on.

We’ll consider the possible outcomes for each round of drawing:

1. **Clark wins on his first turn**: The probability that Clark draws the fruit chew on the first turn is \(\frac{1}{5}\), because there is 1 fruit chew and 5 total treats.

2. **Chloe draws second**: If Clark doesn't win on the first turn (i.e., Clark draws a chocolate), then Chloe has a chance to draw the fruit chew. The probability that Chloe draws the fruit chew on her first turn is \(\frac{1}{4}\), since 1 fruit chew remains out of the 4 remaining treats.

3. **Clark draws third**: If both Clark and Chloe draw chocolates on their first turns, then Clark gets another chance to draw the fruit chew. At this point, there are 3 treats left, and the probability that Clark draws the fruit chew is \(\frac{1}{3}\).

4. **Chloe draws fourth**: If neither Clark nor Chloe has drawn the fruit chew in the first three rounds, then Chloe has a \(\frac{1}{2}\) chance of drawing it on her second turn.

5. **Clark draws fifth**: If the fruit chew hasn't been drawn in the first four turns, then Clark is guaranteed to draw it on his third turn (since there is only one treat left).

### Probability calculation:

We want to calculate the probability that **Clark wins**. Clark wins if:
- He draws the fruit chew on the first or third turn, or on the fifth turn (if it reaches that far).

Let’s calculate this probability step-by-step:

1. The probability that Clark wins on the **first** draw is \(\frac{1}{5}\).
   
2. If Clark doesn't win on the first turn, then the probability that Chloe draws a chocolate on her first turn is \(\frac{4}{5} \times \frac{3}{4} = \frac{12}{20} = \frac{3}{5}\). Then, Clark gets a second chance on the third turn. The probability that he wins on the **third** turn is \(\frac{3}{5} \times \frac{1}{3} = \frac{1}{5}\).

3. If the game reaches the **fifth** turn, Clark is guaranteed to win because there will only be one treat left (the fruit chew). The probability that the game reaches the fifth turn without either Clark or Chloe drawing the fruit chew on their first four turns is \(\frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} = \frac{1}{5}\).

### Total Probability:

The total probability that Clark wins is the sum of these probabilities:
\[
P(\text{Clark wins}) = \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{3}{5}
\]

Thus, the probability that Clark wins is \(\boxed{\frac{3}{5}}\).

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Let me know if you'd like further clarification or details! Here are five related questions to deepen your understanding:

1. What is the probability that Chloe wins?
2. How would the probability change if Chloe were to draw first instead?
3. How does the probability change if the number of chocolates increases to 5 instead of 4?
4. What is the probability that the game reaches the fifth round of drawing?
5. How would you calculate the probability if the treats were replaced after each draw?

**Tip:** When solving probability problems, it helps to break down the situation into distinct cases or outcomes and compute the likelihood for each step.

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 Clark wins?

Calculate the probability that Clark wins in a game where he and Chloe take turns drawing from a box containing 4 chocolates and 1 fruit chew. A step-by-step solution explores probability theory, covering key concepts such as conditional probability and the law of total probability.

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