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 determine the probability that Chloe wins by drawing the fruit chew. This scenario involves drawing treats without replacement, meaning the treats cannot be returned to the box once drawn.

### Step 1: Analyze the problem
- There are 4 chocolates and 1 fruit chew in the box.
- Clark draws first, followed by Chloe, alternating turns.
- The process continues until one of them draws the fruit chew.
- Chloe wins if Clark doesn't draw the fruit chew on his first turn, and Chloe draws it on her turn or later.

### Step 2: Total possible draws
The total number of draws is 5 (4 chocolates + 1 fruit chew).

### Step 3: Cases where Chloe wins
To find the probability that Chloe wins, we consider two possible cases where Chloe could draw the fruit chew:

#### Case 1: Chloe draws the fruit chew on her first turn (second draw overall)
- Clark must not draw the fruit chew on his first draw.
- The probability of Clark drawing a chocolate on his first turn is:
  \[
  P(\text{Clark avoids fruit chew on 1st draw}) = \frac{4}{5}
  \]
- Now, Chloe has a chance to draw the fruit chew on her first turn (the second draw overall). After Clark's draw, there are 4 remaining treats (3 chocolates and 1 fruit chew). The probability that Chloe draws the fruit chew is:
  \[
  P(\text{Chloe draws fruit chew on 2nd draw}) = \frac{1}{4}
  \]
- The combined probability for this case is:
  \[
  P(\text{Chloe wins on 2nd draw}) = \frac{4}{5} \times \frac{1}{4} = \frac{1}{5}
  \]

#### Case 2: Chloe draws the fruit chew on her second turn (fourth draw overall)
- Clark must avoid the fruit chew on his first and third draws.
- The probability of Clark avoiding the fruit chew on his first draw is:
  \[
  P(\text{Clark avoids fruit chew on 1st draw}) = \frac{4}{5}
  \]
- Then, Chloe must draw a chocolate on her first draw (the second draw overall). After Clark's first draw, there are 4 remaining treats, and 3 of them are chocolates. The probability that Chloe draws a chocolate on her first draw is:
  \[
  P(\text{Chloe draws chocolate on 2nd draw}) = \frac{3}{4}
  \]
- Now, Clark must avoid the fruit chew again on his second draw (third draw overall). After the first two draws, 3 treats remain (2 chocolates and 1 fruit chew). The probability that Clark draws a chocolate on his second draw is:
  \[
  P(\text{Clark avoids fruit chew on 3rd draw}) = \frac{2}{3}
  \]
- Finally, Chloe has a chance to draw the fruit chew on her second turn (the fourth draw overall). At this point, there are only 2 treats left, 1 chocolate and 1 fruit chew, so the probability that Chloe draws the fruit chew is:
  \[
  P(\text{Chloe draws fruit chew on 4th draw}) = \frac{1}{2}
  \]
- The combined probability for this case is:
  \[
  P(\text{Chloe wins on 4th draw}) = \frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} = \frac{1}{5}
  \]

### Step 4: Total probability of Chloe winning
Now, we sum the probabilities from the two cases:
\[
P(\text{Chloe wins}) = P(\text{Chloe wins on 2nd draw}) + P(\text{Chloe wins on 4th draw}) = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}
\]

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

Would you like more details on any part of the explanation?

### Related questions:
1. What is the probability that Clark wins?
2. How would the probability change if Chloe draws first?
3. What is the probability that the game ends after exactly 3 draws?
4. How does the probability change if there are 2 fruit chews and 3 chocolates?
5. What is the expected number of draws before someone wins?

### Tip:
When dealing with problems involving turns and alternation, break the problem into individual cases based on possible outcomes after each player's turn.

Explore the probability that Chloe wins in a turn-based drawing game where she and Clark take turns drawing from a box containing 4 chocolates and 1 fruit chew without replacement. Learn how to calculate probabilities step-by-step, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation