Math Problem Statement

Based on a survey, assume that 29% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when six consumers are randomly selected, exactly three of them are comfortable with delivery by drones. Identify the values of n, x, p, and q. The value of n is (Type an integer or a decimal. Do not round.)

Solution

This problem is related to the binomial distribution, where we want to find the probability of a specific number of successes (in this case, consumers comfortable with drones) in a fixed number of trials.

In the binomial distribution, the parameters are defined as follows:

  • nn = number of trials (number of consumers selected)
  • xx = number of successes (number of consumers comfortable with drones)
  • pp = probability of success (probability a consumer is comfortable with drones)
  • qq = probability of failure (probability a consumer is not comfortable with drones), where q=1pq = 1 - p

Now, we can define these values based on the problem:

  • nn = the number of consumers selected = 6
  • xx = the number of successes = 3
  • pp = the probability of success = 0.29 (29% of consumers are comfortable)
  • qq = the probability of failure = 1p=10.29=0.711 - p = 1 - 0.29 = 0.71

Thus, the value of nn is 6.

Would you like to go through the steps to find the probability next?

Here are five related questions:

  1. How do we calculate the binomial probability using the values of nn, xx, pp, and qq?
  2. What is the formula for the binomial probability distribution?
  3. How does increasing the number of trials nn affect the probability distribution?
  4. Can the value of pp in binomial distribution exceed 1?
  5. How does the value of qq relate to pp in this context?

Tip: The sum of the probabilities pp and qq in a binomial distribution always equals 1.

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Math Problem Analysis

Mathematical Concepts

Binomial Distribution
Probability
Combinatorics

Formulas

n = number of trials
x = number of successes
p = probability of success
q = 1 - p

Theorems

Binomial Distribution Theorem

Suitable Grade Level

Grades 10-12