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!