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Chapter 3: Problem 93

Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is \(0.10,\) and the passengers behave independently. (a) What is the probability that every passenger who shows up can take the flight? (b) What is the probability that the flight departs with empty seats?

Short Answer

Expert verified
(a) Calculate \(P(X \leq 120)\). (b) Calculate \(P(X < 120)\).

Step by step solution

01

Define the Random Variable

Let the random variable \( X \) represent the number of passengers who show up for the flight. We want to consider each passenger independently to determine if they show up, with probability \( p = 0.90 \) (since 0.10 is the probability of not showing up). Thus, \( X \) follows a binomial distribution \( \text{Binomial}(n, p) \) where \( n = 125 \) and \( p = 0.90 \).
02

Calculate the Probability for Question (a)

We need to determine the probability that every passenger who shows up can take the flight, which means at most 120 passengers show up out of 125. This can be calculated using the cumulative binomial probability: \[ P(X \leq 120) = \sum_{k=0}^{120} \binom{125}{k} (0.90)^k (0.10)^{125-k} \] This probability needs to be computed using a calculator or software with cumulative binomial functions.
03

Calculate the Probability for Question (b)

To find the probability that the flight departs with empty seats, we must determine the situation when fewer than 120 passengers arrive. This is represented by the probability \[ P(X < 120) = \sum_{k=0}^{119} \binom{125}{k} (0.90)^k (0.10)^{125-k} \] This also requires the use of a calculator or software that computes cumulative binomial probabilities.

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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 a branch of mathematics concerned with the analysis of random phenomena. It is the foundation for understanding and working with uncertainty and randomness, which is a pivotal part of many real-world settings.
In this particular exercise, probability theory helps us evaluate the likelihood of different scenarios when airline passengers show up for their flights. Here, we use concepts from probability theory to calculate probabilities for certain events, like the total number of passengers who show up being less than or equal to the plane's capacity.
In a scenario where each event (each passenger showing up) is independent, we can model these events using a binomial distribution. This forms a principal part of probability theory that deals with random events that have two possible outcomes, such as "show up" or "not show up".
Cumulative Probability
Cumulative probability is used to determine the probability that a random variable is less than or equal to a certain value. In simpler terms, it鈥檚 a way to find out the chance that something occurs up to a specific point.
In the airline problem, cumulative probability is applied by calculating the likelihood that up to 120 passengers show up, using the formula for cumulative distribution of a binomial random variable:
\[ P(X \leq 120) = \sum_{k=0}^{120} \binom{125}{k} (0.90)^k (0.10)^{125-k} \]
This formula sums up the probabilities of all possibilities from 0 passengers up to 120 passengers showing up. It gives us the overall chance that not more than 120 passengers will arrive.
Understanding cumulative probability is crucial for evaluating risks and outcomes in business, science, and daily life scenarios.
Random Variables
Random variables are essential concepts in probability theory. They are variables whose values depend on the outcomes of a random phenomenon. In simple terms, they are used to quantify random processes and are usually denoted by symbols, such as \( X \).
In the context of this exercise, \( X \) represents the number of passengers who show up for the flight. The value of \( X \) can vary depending on how many passengers decide to show up. Here, \( X \) follows a binomial distribution because it counts the number of successes (passengers showing up) in a fixed number of trials (tickets sold), each with the same probability of success (\( p = 0.90 \)).
Random variables like \( X \) enable us to model and make sense of uncertainties, allowing businesses and individuals to plan for different possible outcomes.

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Most popular questions from this chapter

A player of a video game is confronted with a series of opponents and has an \(80 \%\) probability of defeating each one. Success with any opponent is independent of previous encounters. The player continues to contest opponents until defeated. (a) What is the probability mass function of the number of opponents contested in a game? (b) What is the probability that a player defeats at least two opponents in a game? (c) What is the expected number of opponents contested in a game? (d) What is the probability that a player contests four or more opponents in a game? (e) What is the expected number of game plays until a player contests four or more opponents?

The probability that your call to a service line is answered in less than 30 seconds is \(0.75 .\) Assume that your calls are independent. (a) If you call 10 times, what is the probability that exactly nine of your calls are answered within 30 seconds? (b) If you call 20 times, what is the probability that at least 16 calls are answered in less than 30 seconds? (c) If you call 20 times, what is the mean number of calls that are answered in less than 30 seconds?

For each scenario described below, state whether or not the binomial distribution is a reasonable model for the random variable and why. State any assumptions you make. (a) A production process produces thousands of temperature transducers. Let \(X\) denote the number of nonconforming transducers in a sample of size 30 selected at random from the process. (b) From a batch of 50 temperature transducers, a sample of size 30 is selected without replacement. Let \(X\) denote the number of nonconforming transducers in the sample. (c) Four identical electronic components are wired to a controller that can switch from a failed component to one of the remaining spares. Let \(X\) denote the number of components that have failed after a specified period of operation. (d) Let \(X\) denote the number of accidents that occur along the federal highways in Arizona during a one-month period. (e) Let \(X\) denote the number of correct answers by a student taking a multiple-choice exam in which a student can eliminate some of the choices as being incorrect in some questions and all of the incorrect choices in other questions. (f) Defects occur randomly over the surface of a semiconductor chip. However, only \(80 \%\) of defects can be found by testing. A sample of 40 chips with one defect each is tested. Let \(X\) denote the number of chips in which the test finds a defect. (g) Reconsider the situation in part (f). Now, suppose the sample of 40 chips consists of chips with 1 and with 0 defects. (h) A filling operation attempts to fill detergent packages to the advertised weight. Let \(X\) denote the number of detergent packages that are underfilled. (i) Errors in a digital communication channel occur in bursts that affect several consecutive bits. Let \(X\) denote the number of bits in error in a transmission of 100,000 bits. (j) Let \(X\) denote the number of surface flaws in a large coil of galvanized steel.

Data from www.centralhudsonlabs determined the mean number of insect fragments in 225 -gram chocolate bars was \(14.4,\) but three brands had insect contamination more than twice the average. See the U.S. Food and Drug Administration-Center for Food Safety and Applied Nutrition for Defect Action Levels for food products. Assume the number of fragments (contaminants) follows a Poisson distribution. (a) If you consume a 225-gram bar from a brand at the mean contamination level, what is the probability of no insect contaminants? (b) Suppose you consume a bar that is one-fifth the size tested (45 grams) from a brand at the mean contamination level. What is the probability of no insect contaminants? (c) If you consume seven 28.35-gram (one-ounce) bars this week from a brand at the mean contamination level, what is the probability that you consume one or more insect fragments in more than one bar? (d) Is the probability of contamination more than twice the mean of 14.4 unusual, or can it be considered typical variation? Explain.

Suppose that \(X\) has a discrete uniform distribution on the integers 0 through \(9 .\) Determine the mean, variance, and standard deviation of the random variable \(Y=5 X\) and com- pare to the corresponding results for \(X\).
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