91Ó°ÊÓ

In statistics, a binomial distribution describes the number of successes in a fixed number of independent trials, with each trial having the same probability of success. It is widely used to model scenarios where there are two possible outcomes, often termed as 'success' and 'failure.' For example, in our exercise, 'success' refers to a consumer being comfortable with drone deliveries, which has a probability of 0.42.
When working with binomial distributions, it's important to understand the basic properties:

• Fixed number of trials (n): In our problem, n = 5.
• Two possible outcomes: Comfortable (success) or not comfortable (failure).
• Probability of success (p): p = 0.42 for being comfortable.
• Constant probability of success: Each consumer has the same probability of being comfortable with drones.

Utilizing the binomial distribution helps us calculate probabilities for different numbers of successes (k) in n trials.

The combination formula, denoted as \( C(n, k) \), is used to determine the number of ways we can choose k successes from n trials. It is an essential part of the binomial probability formula.
The formula is expressed as:
\[ C(n, k) = \frac{n!}{k!(n - k)!} \]
Where:

• \( n! \): Factorial of n, which is the product of all positive integers up to n.
• \( k! \): Factorial of k.
• \( (n - k)! \): Factorial of (n - k).

In our problem, we need to find the number of ways to choose 2 comfortable consumers out of 5. Thus:
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = 10 \]
This tells us there are 10 different ways to choose which 2 of the 5 consumers are comfortable with drones.

To find the probability of a specific number of successes in a binomial distribution, we use the binomial probability formula:
\[ P(X = k) = C(n, k) \times p^k \times (1 - p)^{n - k} \]
Where:

• \( P(X = k) \): Probability of having exactly k successes.
• \( C(n, k) \): Number of combinations of n items taken k at a time.
• \( p \): Probability of success.
• \( (1-p) \): Probability of failure.

For our exercise, we want to find the probability that exactly 2 out of 5 consumers are comfortable with drones:
\[ P(X = 2) = C(5, 2) \times (0.42)^2 \times (0.58)^3 \]
We first calculate the individual parts:
\( C(5, 2) = 10 \) (as calculated earlier), \( (0.42)^2 = 0.1764 \), and \( (0.58)^3 = 0.195112 \).
Therefore:
\[ P(X = 2) = 10 \times 0.1764 \times 0.195112 = 10 \times 0.0344099648 = 0.344099648 \approx 0.34410 \]
This calculation shows the correct probability of selecting exactly 2 comfortable consumers out of 5.

In the initial approach, a multiplication rule was mistakenly used. This rule means multiplying the probabilities of individual outcomes to get the overall probability. It's suitable for dependent or sequential events but not for binomial distributions.
The error in our exercise arose because it only considered a single sequence (2 comfortable followed by 3 not comfortable) without acknowledging that these events can occur in multiple orders. Thus,
\[ (0.42)(0.42)(0.58)(0.58)(0.58)=0.0344 \]
is incorrect because it captures only one sequence of the events. The correct approach, using the combination formula, accounts for all possible sequences where 2 out of 5 are comfortable consumers.
Always apply binomial probability methods when dealing with fixed trials and independent events with two outcomes. This error highlights the necessity to thoroughly understand the differences between approaches for correctly evaluating probabilities in different contexts.