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Chapter 4: Problem 6

Find the indicated complements. According to the Bureau of Transportation, \(80.3 \%\) of American Airlines flights arrive on time. What is the probability of randomly selecting an American Airlines flight that does not arrive on time?

Short Answer

Expert verified
The probability of a flight not arriving on time is 0.197.

Step by step solution

01

Understand the given probability

The problem states that 80.3% of American Airlines flights arrive on time. This percentage is given as the probability of on-time arrivals. Let's denote this probability as P(A) = 0.803.
02

Identify the complement event

The complement of the event of a flight arriving on time is the event of a flight not arriving on time. We want to find the probability of this complement event.
03

Use the complement rule

The probability of the complement of an event is equal to 1 minus the probability of the event. Mathematically, this is expressed as P(A') = 1 - P(A). Substituting the given probability: P(A') = 1 - 0.803.
04

Calculate the complement probability

Perform the arithmetic calculation: P(A') = 1 - 0.803 P(A') = 0.197.

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Key Concepts

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

Events and Complements
In probability, events and complements are essential concepts. An event is an outcome or a specific set of outcomes of a random experiment. For instance, a flight arriving on time is an event. The complement of an event includes all possible outcomes not covered by the original event. If we know an event, like a flight arriving on time, its complement would be the flight not arriving on time. Thus, these two events together cover all possible scenarios in the experiment.
Understanding complements helps in solving probability problems more efficiently, as sometimes it's easier to calculate the probability of the complement and then derive the desired probability.
Probability Calculation
Calculating probabilities is foundational in understanding how likely events are to occur. In our exercise, we know that the probability of an American Airlines flight arriving on time, denoted as P(A), is 0.803.
To find the probability of the complement (flight not arriving on time), we use the basic formula for complements: P(A') = 1 - P(A). Here, P(A') represents the complement probability.
Using the given probability, we calculate:
P(A') = 1 - 0.803 = 0.197.
So, the probability of a flight not arriving on time is 0.197, or 19.7%.
Probability Rules
Several rules govern how probabilities work, making these calculations and understandings possible:
  • Addition Rule: For mutually exclusive events A and B, P(A or B) = P(A) + P(B). This applies when events cannot happen simultaneously, like rolling a die and getting a 1 or 6.
  • Multiplication Rule: For independent events A and B, P(A and B) = P(A) * P(B). This is used when the occurrence of one event does not affect the other, like flipping a coin and rolling a die.
  • Complement Rule: The probability of event A's complement is P(A') = 1 - P(A). This is utilized when you know the probability of an event but want to find out the chance of it not occurring.

These rules help simplify and solve complex probability problems effectively.

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Most popular questions from this chapter

Express all probabilities as fractions. As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 75 and, in a separate drawing, you must also select the correct single number between 1 and 15 . Find the probability of winning the jackpot. How does the result compare to the probability of being struck by lightning in a year, which the National Weather Service estimates to be \(1 / 960,000\) ?

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Express all probabilities as fractions. DJ Marty \(\mathrm{T}\) is hosting a party tonight and has chosen 8 songs for his final set (including "Daydream Believer" by the Monkees). How many different 8 -song playlists are possible (song order matters)? If the 8 songs are randomly selected, what is the probability they are in alphabetical order by song title?
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