I have 46 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?

To find the probability that one of the nine balls you took out is the red ball, we can approach the problem using the concept of combinations.

### Step 1: Determine the total number of possible outcomes
- You have 46 balls in total.
- You are selecting 9 balls from these 46.
- The total number of ways to select 9 balls from 46 is given by the combination formula \( \binom{46}{9} \).

### 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 45 blue balls. The number of ways to do this is given by \( \binom{45}{8} \).

### 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:

\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{\binom{45}{8}}{\binom{46}{9}}
\]

Let's calculate this probability.

\[
P = \frac{\binom{45}{8}}{\binom{46}{9}} 
\]

We can compute this using a calculator:

\[
\binom{45}{8} = \frac{45!}{8!(45-8)!}
\]
\[
\binom{46}{9} = \frac{46!}{9!(46-9)!}
\]

After computing these, we find:

\[
P = \frac{234,086,340}{352,716,552} \approx 0.6638
\]

So, the probability that one of the balls you took out is the red ball is approximately **0.6638**, or **66.38%**.

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 would the probability change if there were more than one red ball?
3. What if you selected 10 balls instead of 9? How does that affect the probability?
4. How would you calculate the probability if you only knew the percentage of red balls?
5. What is the probability of selecting exactly two red balls if you had multiple red balls?

**Tip:** When dealing with probability problems involving combinations, it's helpful to start by clearly defining the total and favorable outcomes.

Calculate the probability of drawing one red ball from a pocket containing 46 balls, where one ball is red and the rest are blue. Understand the concept of combinations and use the binomial coefficient formula to solve the problem. Detailed step-by-step solution provided.

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