In​ February, a major airline had 77.8​% 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.

### Part (a): Probability that all four flights arrive on time

We are given that the probability that a flight arrives on time is 77.8%, or 0.778. The problem assumes that each flight's arrival time is independent of the others. To find the probability that all four flights arrive on time, we can multiply the probabilities of each flight arriving on time, since the events are independent.

The probability of all four flights arriving on time is:

\[
P(\text{all four flights on time}) = 0.778 \times 0.778 \times 0.778 \times 0.778
\]

Alternatively, this can be written as:

\[
P(\text{all four flights on time}) = (0.778)^4
\]

Now, let's calculate this:

\[
P(\text{all four flights on time}) = 0.778^4 \approx 0.365
\]

Thus, the probability that all four flights arrive on time is approximately **0.365**, or **36.5%**.

### Part (b): Discussing the realism of the independence assumption

The assumption that the on-time arrival of one flight is independent of another may not always be realistic. In reality, flights often experience delays due to external factors that can affect multiple flights simultaneously. Some examples include:

- **Weather conditions**: Bad weather at a particular airport could delay multiple flights.
- **Air traffic congestion**: If a major airport faces congestion or operational issues, multiple flights departing from or arriving at that airport could be delayed.
- **Aircraft availability**: If an aircraft is delayed for one flight, it could affect subsequent flights using the same plane.

These common situations suggest that delays on one flight may increase the likelihood of delays on other flights, especially if they are part of a connecting itinerary or share the same origin/destination. Therefore, assuming perfect independence might oversimplify the actual situation. A more realistic approach might account for correlated events, especially in cases of bad weather or system-wide disruptions.

Would you like any further details or clarification on this?

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Here are five related questions to expand your understanding:
1. What is the probability that exactly three out of four flights arrive on time?
2. How would the probability change if the on-time performance increased to 85%?
3. How do dependent events change the overall probability in a situation like this?
4. How would you calculate the expected number of flights to be on time if you take 10 flights?
5. How might connecting flights further complicate the assumption of independence?

**Tip**: Always check if events are truly independent or if external factors could influence multiple outcomes simultaneously. This can significantly affect probability calculations.

In February, a major airline had 77.8% 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.

Find the probability that all four flights arrive on time, assuming a 77.8% on-time performance rate. Learn how to apply the multiplication rule for independent events and discuss the realism of independence assumptions in probability scenarios.

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