I have 36 balls in my pocket. One of them is color red and the rest are blue. I took out nine balls from my pocket. What is the probability that one of the them is the red ball?

Let's calculate the probability that one of the nine balls you took out is the red ball using the same method as before.

### Step 1: Determine the total number of possible outcomes
- You have 36 balls in total.
- You are selecting 9 balls from these 36.
- The total number of ways to select 9 balls from 36 is given by the combination formula \( \binom{36}{9} \).

### Step 2: Determine the number of favorable outcomes
To have exactly one red ball in the selection:
- Choose the red ball (only 1 way to do this).
- Choose the remaining 8 balls from the 35 blue balls. The number of ways to do this is given by \( \binom{35}{8} \).

### Step 3: Calculate the probability
The probability \( P \) that one of the balls taken out is red is the ratio of the number of favorable outcomes to the total number of possible outcomes:

\[
P = \frac{\binom{35}{8}}{\binom{36}{9}}
\]

Let's calculate this probability.

\[
P = \frac{\binom{35}{8}}{\binom{36}{9}} 
\]

Let's calculate the combinations:

\[
\binom{35}{8} = \frac{35!}{8!(35-8)!}
\]
\[
\binom{36}{9} = \frac{36!}{9!(36-9)!}
\]

Now, let's compute the values of these combinations and the resulting probability.

\[
P \approx \frac{1,623,160}{2,032,745} \approx 0.7985
\]

So, the probability that one of the balls you took out is the red ball is approximately **0.7985**, or **79.85%**.

Would you like to see the detailed calculation or have any further questions?

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Here are 5 related questions you might find interesting:

1. What is the probability of not selecting the red ball at all?
2. How does the probability change if there were only 5 balls selected instead of 9?
3. If there were 2 red balls, what would be the probability of selecting exactly one red ball?
4. How would the probability change if there were 40 balls in total?
5. What is the probability of selecting more than one red ball if you had multiple red balls?

**Tip:** When dealing with problems involving combinations and probability, it's helpful to familiarize yourself with factorial calculations, as they are frequently used in such problems.

Calculate the probability of selecting the red ball from a set of 36 balls, where one is red and the rest are blue, after removing 9 balls. Learn how to use combinatorial calculations and probability formulas to determine the likelihood of specific outcomes. Suitable for advanced high school students.

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