Question

A Department of Transportation report about air travel found that airlines misplace about 5 bags per 1000 passengers. Suppose you are traveling with a group of people who have checked 22 pieces of luggage on your flight. Can you consider the fate of these bags to be Bernoulli trials? Explain.

Short answer

Yes, the fate of the luggage can be considered Bernoulli trials.

Step-by-step solution

Identify Bernoulli Trial Criteria

A Bernoulli trial is an experiment or process that meets four criteria: (1) there are only two possible outcomes, (2) each trial is independent, (3) the probability of each outcome is the same for every trial, and (4) the number of trials is fixed.

Determine If There Are Two Possible Outcomes

Check if there are only two outcomes for each piece of luggage. In this case, a piece of luggage can either be misplaced or not be misplaced, satisfying the first criterion of a Bernoulli trial.

Check Independence of Trials

Consider whether the fate of each piece of luggage is independent of the others. Assuming that the misplacement of one bag does not affect the others, this condition is satisfied, meeting the second criterion.

Verify Equal Probability of Each Outcome

Ensure that the probability of a piece of luggage being misplaced is the same for each luggage. It is given that the probability is 5 misplaced bags per 1000 passengers for each piece of luggage, satisfying the third criterion.

Confirm Fixed Number of Trials

There are 22 pieces of luggage, which sets a fixed number of trials. This meets the fourth criterion for Bernoulli trials.

Conclusion on Bernoulli Trials

Since all four criteria for Bernoulli trials are satisfied, the fate of the 22 pieces of luggage can indeed be considered as Bernoulli trials.

Key concepts

Probability

Probability refers to the measure of the likelihood that an event will occur. It can range from 0 (impossible) to 1 (certain). In Bernoulli trials, understanding probability is crucial because each trial has a specific chance of resulting in the "success" or "failure" outcomes.

In the context of the given problem, the probability of a piece of luggage being misplaced is 0.5%. This is computed from the given data that 5 bags are misplaced per 1000 passengers.

When dealing with probability in Bernoulli trials, it is important to consistently apply the same probability to each trial, ensuring uniformity. Thus, even as new luggage is checked in, the likelihood of any single piece being misplaced remains 0.5%.

Independent Trials

For a series of experiments to be classified as Bernoulli trials, each trial must be independent. This means that the outcome of one trial does not influence the outcome of another.

In the luggage example, if the fate of any piece of luggage was influenced by whether other pieces were misplaced or not, the trials would cease to be independent.

• Suppose bag A is misplaced, yet bag B still has the same chance of being misplaced as it initially did.
• Independence ensures that the probability remains constant for each trial regardless of previous outcomes.

Understanding independence helps maintain the integrity of the Bernoulli process by assuring trial outcomes are unaffected by external factors.

Binary Outcomes

Binary outcomes refer to scenarios where there are only two possible results. Bernoulli trials rely on this concept. In our case:

• The luggage can either be misplaced (considered as one outcome).
• Or it can not be misplaced (another distinct outcome).

Such a clear distinction into two possibilities simplifies analysis and calculations.

This duality makes Bernoulli trials powerful, as they allow for the modeling of complex decision-making scenarios with simple "success" or "failure" outcomes.

Fixed Number of Trials

A fixed number of trials is a fundamental aspect of Bernoulli trials. This means the number of experiments or observations is predetermined and does not change.

In our scenario, exactly 22 pieces of luggage have been checked, providing a fixed number of 22 trials. This fixed number ensures the completion of a full "experiment" cycle, allowing for meaningful computations and conclusions.

• Having a fixed number provides closure to, and scope for, the analysis.
• It helps in calculating probabilities and expectations confidently.

The fixed nature simplifies predicting outcomes since the number of trials remains constant throughout the experiment.