Question

in february, a major airline had 77.4% 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 three 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 three 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 $p = 0.774$.

Step2: Use the multiplication rule for independent events

Since the on - time arrivals of the three flights are independent, the probability that all three flights arrive on time is $P=p\times p\times p=p^{3}$.
Substitute $p = 0.774$ into the formula: $P=(0.774)^{3}$.
Calculate $(0.774)^{3}=0.774\times0.774\times0.774 = 0.4645$.

In reality, the assumption of independence of on - time arrivals of different flights may not be entirely accurate. 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 a hub airport can cause delays for many flights departing from or arriving at that airport. So, while the independence assumption simplifies the probability calculation, it does not fully account for real - world interdependencies among flights.

Answer:

$0.4645$