https://file.aiquickdraw.com/mathbot/file/17375268622202znoab4a.jpg

The summation formula shown in the image appears to calculate the expected value of a binomial random variable scaled by \( kx \). Here's the formula:

\[
\sum_{k=0}^{30} k \cdot \binom{30}{k} \cdot \left( \frac{1}{12} \right)^k \cdot \left( \frac{11}{12} \right)^{30-k}.
\]

To solve this:

### Step 1: Recognize this as a property of the binomial distribution.
The formula simplifies using the expectation property of the binomial distribution:
- A binomial random variable \( X \sim B(n, p) \) has \( E[X] = n \cdot p \).

Here:
- \( n = 30 \) (number of trials),
- \( p = \frac{1}{12} \) (probability of success).

### Step 2: Simplify the summation.
Using the expected value property:
\[
E[X] = 30 \cdot \frac{1}{12} = 2.5.
\]

Since \( k \cdot x = k \), the answer is directly the expectation:
\[
2.5
\]

Let me know if you want further details or explanations!

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### Questions:
1. How is the expectation of a binomial distribution derived mathematically?
2. What does the summation property in the binomial distribution indicate?
3. How can this problem be solved computationally for larger values of \( n \)?
4. What are the practical applications of binomial distributions in probability?
5. How does changing \( p \) to other fractions affect the expected value?

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### Tip:
For binomial summations involving \( k \cdot \binom{n}{k} \), use the expectation formula \( E[X] = n \cdot p \) to simplify your calculations!

Learn how to calculate the expected value of a binomial random variable using a summation formula. This problem involves key concepts like probability, the binomial distribution, and expected value, suitable for undergraduate students.

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