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In probability, two events are considered independent if the occurrence of one does not affect the occurrence of the other. In other words, the outcome of one trial does not influence the outcome of another trial. When you select 30 consumers from a group of 1009, you might initially think that since they are chosen without replacement, the trials are not independent. However, in many practical scenarios, we can apply a simplification rule that helps us treat the selections as independent. This brings us to the 5% rule in sampling.

Remember, ensuring independence simplifies calculations and models significantly, making our statistical analysis more straightforward. That’s why determining independence is crucial before proceeding further.

The 5% rule in sampling states that when you are dealing with large populations, a sample can be considered independent if it is less than 5% of the total population, even when sampling without replacement. This is because the changes in probabilities are minimal and can be ignored for practical purposes.

Here's how this works for our problem:

• First, we calculate 5% of the population (1009 consumers).Using the formula \[0.05 \times 1009 = 50.45\].
• Then, we compare this result to our sample size (30 consumers). Since 30 is less than 50.45, we can treat these selections as independent.

With the 5% rule, even though we initially selected without replacement, the negligible effect on probabilities allows the simplification of treating selections as independent.

The binomial probability formula is a key tool in statistics for calculating the probability of a given number of successes in a fixed number of independent trials. It is given by:

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

where:

• \( P(X = k) \) is the probability of getting exactly \( k \) successes in \( n \) trials.
• \( n \) is the number of trials.
• \( k \) is the number of successes desired.
• \( p \) is the probability of success on a single trial.

For our problem, we need to check whether this formula applies:

• Are the trials independent? Using the 5% rule, we treated them as independent.
• Are there only two possible outcomes (success or failure)? Yes, a consumer is either comfortable with drones (success) or not (failure).

Thus, by confirming these conditions, we can use the binomial probability formula to calculate the probability that at least 10 out of the 30 consumers are comfortable with drones from the given survey sample.