Question

in february, a major airline had 77.9% of their flights arrive on time. assume that the event that a given flight arrives on time is independent of the event that another flight arrives on time.
a. a writer plans to take four separate flights for her publisher next month. assuming the airline has the same on - time performance as in february, what is the probability that all four flights arrive on time?
b. discuss how realistic it is to assume that the on - time arrivals of the different flights are independent.
a. the probability is
(simplify your answer. round to four decimal places as needed.)

Explanation:

Step1: Identify the probability of a single - flight on - time arrival

The probability that a single flight arrives on time is $p = 0.779$.

Step2: Use the multiplication rule for independent events

Since the on - time arrivals of the four flights are independent, the probability that all four flights arrive on time is $P=p\times p\times p\times p=p^{4}$.
Substitute $p = 0.779$ into the formula: $P=(0.779)^{4}$.
$(0.779)^{4}=0.779\times0.779\times0.779\times0.779 = 0.3679$.

Step3: Discuss the independence assumption

In reality, the on - time arrivals of different flights may not be completely independent. For example, factors such as weather conditions at the departure or arrival airports, air traffic control issues in a particular region, and mechanical problems with the aircraft fleet can affect multiple flights simultaneously. If there is a snowstorm at the airport, it can cause delays for many flights. Also, if there are air traffic congestion issues in a major air - traffic hub, it can impact the on - time performance of several flights passing through that area. So, the assumption of independence may be too simplistic in practice.

Answer:

a. $0.3679$
b. The assumption that the on - time arrivals of different flights are independent may be unrealistic because factors like weather, air traffic control, and mechanical problems can affect multiple flights simultaneously.