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In the context of the binomial distribution, probability calculation involves finding the likelihood that a certain number of trials result in success, which in our case is a child having a food allergy. This is done using the binomial probability formula:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:

• \( n \) is the total number of trials (children sampled, 25 in our case),
• \( k \) is the number of successful trials (children with allergies),
• \( p \) is the probability of success on a single trial (0.05 for a food allergy),
• \( \binom{n}{k} \) is a binomial coefficient representing the number of ways to choose \( k \) successes from \( n \) trials.

For this exercise, you'll calculate probabilities for various values of \( X \) using this formula. For instance, finding \( P(X \leq 3) \) requires summing the individual probabilities for \( X = 0, 1, 2, \) and \( 3 \). It's important to carefully compute each part of the formula to ensure correct results.

The expected value in a binomial distribution gives us the average number of successes we'd expect from a large number of trials. It's a central measure that helps predict future outcomes based on the distribution's parameters.

The expected value \( E(X) \) of a binomial distribution is calculated using:
\[ E(X) = np \]
In our exercise, the expected value can be found by multiplying the number of trials \( n = 25 \) by the probability of success \( p = 0.05 \), which results in:
\[ E(X) = 25 \times 0.05 = 1.25 \]
This means that, on average, out of the 25 children sampled, we expect about 1.25 children to have a food allergy. Even though practically, a fraction of a child doesn't make sense, this calculation helps guide expectations over many repeated samples.

Standard deviation in a binomial distribution tells us how much the number of successes (children with food allergies) is expected to vary from the average (expected value). It's a key component in understanding the distribution's spread or variability.

To find the standard deviation \( \sigma_X \) for a binomial distribution, use the formula:
\[ \sigma_X = \sqrt{np(1-p)} \]
For our case, plug in the values: \( n = 25 \), \( p = 0.05 \), and \( 1-p = 0.95 \).
Thus, the calculation is:
\[ \sigma_X = \sqrt{25 \times 0.05 \times 0.95} \]
Calculating this gives us a standard deviation, indicating the extent of variation from the expected value of approximately 1.25 children with allergies in our sample.

The complement rule is an essential concept in probability that helps when it's easier to calculate the probability of the opposite (complementary) event, especially when you want probabilities like "at least" or "at most".

If you want to find the probability of at least 4 children having a food allergy (\( P(X \geq 4) \)), you can use the complement rule.

The formula using the complement rule is:
\[ P(X \geq 4) = 1 - P(X \leq 3) \]
Here, \( P(X \leq 3) \) can be calculated as the sum of probabilities for \( X = 0, 1, 2, \) and \( 3 \). Once you have \( P(X \leq 3) \), subtract it from 1 to find \( P(X \geq 4) \).

This approach simplifies computations by focusing on the probabilities of fewer outcomes, effectively making precise calculations more accessible.