Math Problem Statement

Based on a​ survey, assume that

4343​%

of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when

sixsix

consumers are randomly​ selected, exactly

twotwo

of them are comfortable with delivery by drones. Identify the values of​ n, x,​ p, and q.

Solution

In this problem, you're working with a binomial probability distribution, which is used to model the number of successes in a fixed number of independent trials when each trial has the same probability of success.

The key values you need to identify are:

  • nn: The number of trials (in this case, the number of consumers surveyed).
  • xx: The number of successes (consumers who are comfortable with drones delivering their purchases).
  • pp: The probability of success on a single trial (the percentage of consumers who are comfortable with drones).
  • qq: The probability of failure on a single trial (the percentage of consumers who are not comfortable with drones).

From the problem:

  1. n=6n = 6 (the number of consumers randomly selected).
  2. x=2x = 2 (we are interested in finding the probability that exactly two consumers are comfortable).
  3. p=43100=0.43p = \frac{43}{100} = 0.43 (since 43% of consumers are comfortable with drone deliveries).
  4. q=1p=10.43=0.57q = 1 - p = 1 - 0.43 = 0.57 (since the remaining percentage, 57%, are not comfortable with drone deliveries).

Recap:

  • n=6n = 6
  • x=2x = 2
  • p=0.43p = 0.43
  • q=0.57q = 0.57

Would you like further details on how to calculate the binomial probability using these values?

Five related questions:

  1. How do you compute binomial probabilities?
  2. What are the properties of a binomial distribution?
  3. How do you calculate the mean and standard deviation of a binomial distribution?
  4. What happens if you change the value of xx in this problem?
  5. What is the difference between binomial and normal distributions?

Tip:

In a binomial distribution, as the number of trials nn increases, the distribution tends to look more like a normal distribution, especially when pp is not too close to 0 or 1.

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Math Problem Analysis

Mathematical Concepts

Probability
Binomial Distribution

Formulas

Binomial Probability Formula: P(X = x) = C(n, x) * p^x * q^(n - x)
C(n, x) = n! / (x! * (n - x)!)

Theorems

Binomial Theorem
Complementary Rule in Probability

Suitable Grade Level

Grades 9-12