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 arrival

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

Step2: Use the multiplication rule for independent events

Since the flights are independent, the probability that all $n = 4$ flights arrive on time is given by $P(X = 4)=p\times p\times p\times p=p^{n}$. Substitute $p = 0.777$ and $n = 4$ into the formula: $P(X = 4)=(0.777)^{4}$.

Step3: Calculate the result

$(0.777)^{4}=0.777\times0.777\times0.777\times0.777\approx0.3669$.

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 an airport can cause delays for many flights departing or arriving from that airport. Also, if an airline has a limited number of available aircraft or crew, a delay in one flight can have a cascading effect on subsequent flights.

Answer:

$0.3669$