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Chapter 7: Problem 14

Of all airline flight requests received by a certain discount ticket broker, \(70 \%\) are for domestic travel (D) and \(30 \%\) are for international flights (I). Let \(x\) be the number of requests among the next three requests received that are for domestic flights. Assuming independence of successive requests, determine the probability distribution of \(x\). (Hint: One possible outcome is DID, with the probability \((.7)(.3)(.7)=.147 .\) )

Short Answer

Expert verified
The probability distribution of \(x\) described in the problem is \( P(x) \), where \(P(0) = 0.027, P(1) = 0.189, P(2) = 0.441, P(3) = 0.343\). So, the probability of getting 0, 1, 2, or 3 requests for domestic flights among the next 3 requests received is 0.027, 0.189, 0.441, and 0.343 respectively.

Step by step solution

01

Identify the probability of success and failure

We are given that the probability of receiving a request for a domestic flight (D) is \(70 \%\) or \(0.7\) and for an international flight (I) is \(30 \%\) or \(0.3\). These are our probabilities of success and failure respectively.
02

Calculate the probabilities for each possible outcome

We are asked to find the probability distribution of \(x\), where \(x\) is the number of successes (domestic flights) in three trials. The possible values for \(x\) are 0, 1, 2, 3. We use the formula for the binomial probability which is \(P(x) = {n \choose x} * (p^x) * (q^{n-x})\) where \(n=3\) is the number of trials, \(x\) is the number of successes, \(p=0.7\) is the probability of success, \(q=0.3\) is the probability of failure, and \( {n \choose x} \) is a binomial coefficient representing the number of ways we can choose \(x\) successes from \(n\) trials.
03

Calculate binomial coefficient

The binomial coefficient is calculated as \({n \choose x} = \frac{n!}{x!(n-x)!}\) where \(n!\) denotes the factorial of number \(n\) which is the product of all positive integers up to \(n\).
04

Generate the probability distribution

Calculate the probability for each possible outcome \(x = 0, 1, 2, 3\). Substituting all values into the binomial formula provides the probability distribution of \(x\).

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Key Concepts

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

Binomial Probability
Binomial probability is a fundamental concept in statistics and probability theory. It deals with scenarios where there are two possible outcomes for each trial 鈥� typically termed success and failure. In our exercise, the success is receiving a request for a domestic flight, and failure is receiving a request for an international one. The formula for binomial probability is:
  • \( P(x) = {n \choose x} \cdot (p^x) \cdot (q^{n-x}) \)
In this context, \( P(x) \) represents the probability of obtaining exactly \( x \) successes in \( n \) trials. Here:
  • \( n = 3 \): Total flight requests considered.
  • \( p = 0.7 \): Probability of a domestic flight request (success).
  • \( q = 0.3 \): Probability of an international flight request (failure).
The binomial probability helps us determine how likely it is to get a certain number of domestic flight requests out of the three.
Binomial Coefficient
The binomial coefficient, often denoted as \( {n \choose x} \), plays a crucial role in calculating the probabilities in binomial distributions. The binomial coefficient can be calculated using the formula:
  • \( {n \choose x} = \frac{n!}{x! (n-x)!} \)
In our exercise, it represents the number of ways to choose \( x \) domestic flight requests (successes) from \( n = 3 \) total requests.The factorial notation \( n! \) ("n factorial") means multiplying all whole numbers from \( n \) down to 1. For example, for \( n = 3 \) and \( x = 0, 1, 2, 3 \), the binomial coefficients are:
  • \( {3 \choose 0} = 1 \)
  • \( {3 \choose 1} = 3 \)
  • \( {3 \choose 2} = 3 \)
  • \( {3 \choose 3} = 1 \)
These coefficients show the number of different ways we can achieve each number of successes out of our three trials.
Probability of Success and Failure
Probability of success and failure forms the basis for calculating binomial probabilities. In our exercise, these probabilities are already defined:
  • Probability of success (domestic flight request): \( p = 0.7 \).
  • Probability of failure (international flight request): \( q = 0.3 \).
Together, they explain the likelihood of each trial outcome within the sequence of requests. Since each flight request is independent, the overall probability for any sequence of requests is a result of multiplying the probabilities for each request type (as shown in DID, where \( (.7)(.3)(.7) = .147 \)).Understanding the probability of both success and failure is essential when using the binomial formula, as it helps signify how these rates influence the final probability distribution over multiple trials. These probabilities always sum up to 1, as every request is either for a domestic or international flight.

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