Four balls are drawn from box containing 2 red, 2 white and 1 yellow balls. What is the probability that the remaining ball is red?

Let's break down the problem step by step:

### Given:
- The box contains:  
  2 red balls  
  2 white balls  
  1 yellow ball  

Thus, there are a total of 5 balls in the box.

We are drawing 4 balls, and we want to calculate the probability that the remaining ball is red.

### Total possible outcomes:
The total number of ways to draw 4 balls out of 5 is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of balls and \( k \) is the number of balls drawn:

\[
\text{Total number of outcomes} = \binom{5}{4} = 5
\]

### Favorable outcomes:
To ensure that the remaining ball is red, one of the two red balls must be left behind, meaning only one red ball is drawn among the 4 drawn balls. The other 3 drawn balls will come from the 2 white and 1 yellow balls. 

So, we must choose 1 red ball from the 2 red balls, and 3 balls from the remaining 3 (2 white + 1 yellow). The number of favorable outcomes is:

\[
\text{Favorable outcomes} = \binom{2}{1} \times \binom{3}{3} = 2 \times 1 = 2
\]

### Probability:
The probability is the ratio of favorable outcomes to total outcomes:

\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{2}{5}
\]

### Conclusion:
The probability that the remaining ball is red is \( \frac{2}{5} \).

Would you like more details on any of these steps, or do you have further questions?

Here are 5 related questions to consider:
1. How would the probability change if the box had more balls of each color?
2. What if you were asked for the probability of drawing exactly 1 red ball?
3. How can this problem be modeled using conditional probability?
4. How do combinations work in probability problems like this one?
5. What if 3 balls were drawn instead of 4? How would the probability change?

**Tip:** In probability, breaking the problem into total outcomes and favorable outcomes helps simplify complex questions.

Four balls are drawn from a box containing 2 red, 2 white, and 1 yellow balls. What is the probability that the remaining ball is red?

Explore the probability of drawing 4 balls from a box containing 2 red, 2 white, and 1 yellow ball, with the goal of determining the likelihood that the remaining ball is red. This detailed solution covers key concepts in probability and combinatorics, ideal for students in Grades 9-12.

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