Question

in february, a major airline had 77.7% 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

The probability that a single flight arrives on time is $p = 0.777$.

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 $p = 0.777$ into the formula: $P=(0.777)^{4}$.
Calculate $(0.777)^{4}=0.777\times0.777\times0.777\times0.777\approx0.3670$.

In reality, the on - time arrivals of different flights may not be truly independent. Factors such as weather conditions at the departure and arrival airports, air traffic control issues, and mechanical problems with the aircraft can affect multiple flights simultaneously. For example, a major snowstorm at an airport can delay or cancel many flights, meaning that if one flight is delayed, others are likely to be affected too. Also, if there is a mechanical issue with a particular type of aircraft and the airline has multiple flights using that type, it can impact multiple flights' on - time performance.

Answer:

$0.3670$