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Chapter 3: Problem 209

Asupplier of heavy construction equipment has found that new customers are normally obtained through customer requests for a sales call and that the probability of a sale of a particular piece of equipment is .3. If the supplier has three pieces of the equipment available for sale, what is the probability that it will take fewer than five customer contacts to clear the inventory?

Short Answer

Expert verified
The probability is approximately 8.37%.

Step by step solution

01

Understand the Problem

We aim to determine the probability that fewer than five customer contacts are required to sell three pieces of equipment, where the probability of selling one piece per contact is 0.3.
02

Identify the Distribution

This situation involves a binomial process where the probability of a sale (success) is 0.3 for each contact, and we want to find the probability that three sales (successes) occur in fewer than five contacts.
03

Define the Random Variable and Criteria

Let X denote the number of contacts needed to make three sales. We are looking for \( P(X < 5) \).
04

Use the Negative Binomial Distribution

Because we want the trials to continue until a fixed number of successes (sales) is achieved, the scenario follows a negative binomial distribution.
05

Calculate the Probability

The probability that fewer than 5 contacts are needed for 3 sales can be calculated by finding \( P(X = 3) + P(X = 4) \). The formula for the negative binomial distribution is:\[ P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r} \]where \( r = 3 \), \( p = 0.3 \), and \( k \) ranges over possible trials (3 and 4 in this case).Calculate \( P(X = 3) \): \[ P(X = 3) = \binom{2}{2} (0.3)^3 (0.7)^0 = 0.027 \]Calculate \( P(X = 4) \): \[ P(X = 4) = \binom{3}{2} (0.3)^3 (0.7)^1 = 3 \cdot 0.027 \cdot 0.7 = 0.0567 \]Add the probabilities: \[ P(X < 5) = P(X = 3) + P(X = 4) = 0.027 + 0.0567 = 0.0837 \]
06

Finalize the Result

The result shows that the probability of needing fewer than five contacts to sell all three pieces of equipment is approximately 0.0837 or 8.37%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It provides tools to model and understand random events in a structured way. In the context of the problem, probability theory helps us calculate the chance of selling equipment within a certain number of customer contacts.

A key aspect of probability theory is understanding events, which are possible outcomes of a situation. For example:
  • The probability of a sale occurring during a contact is 0.3.
  • This value of 0.3, termed as success probability, is constant across contacts.
Probability theory uses mathematical expressions to measure these likelihoods formally. In practice, calculations like the ones in the given problem rely heavily on probability theory principles.
Binomial Process
The binomial process is a statistical model used to describe experiments with a fixed number of trials, two possible outcomes per trial (success or failure), and a constant probability of success. In simpler terms, it's like flipping a coin a set number of times to see how many times it lands heads up.

In the given exercise, each customer contact is a trial, and a sale is considered a success. The scenario presented isn't a perfect binomial process but a variation known as the negative binomial distribution.

Key features:
  • Trials: Each customer contact is a single trial.
  • Outcome: Each contact can result in either a sale (success) or no sale (failure).
  • Probability: The probability of success (sale per contact) is 0.3.
Understanding these elements helps to apply the concept accurately to calculate probabilities.
Random Variable
A random variable is a quantitative description of the outcomes of a random process. In probability theory, it assigns numerical outcomes to random events. It helps to translate real-world scenarios into a structured mathematical form that is easier to analyze.

In the problem, the random variable is "X," which represents the number of customer contacts needed to achieve three sales.

Considerations:
  • Discrete Random Variable: "X" takes integer values, representing countable outcomes in this setup.
  • Probabilistic Behaviour: The distribution of "X" helps determine probabilities for various outcomes, like making sales within a specific number of contacts.
Grappling with the concept of random variables clarifies how probabilities are systematically assigned to specific scenarios, leading to a better understanding of statistical problems.

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