The bureau of Transportation Statistics reports on-time performance for airlines at major u.S. airports. Jetblue, united, and uS Airways share terminal C at boston’s logan Airport. The percentage of on-time flights reported for August 2012 was 76.8% for Jetblue, 71.5% for united, and 82.2% for uS Airways ( bureau of Transportation Statistics website, october 2012). Assume that 30% of the flights arriving at terminal C are Jetblue flights, 32% are united flights, and 38% are uS Airways flights. A. What is the most likely airline for Flight 1382? What is the probability that Flight 1382 is by this airline? B. Suppose that an announcement is made saying that Flight 1382 will now be arriving late. What is the most likely airline for this flight? What is the probability that Flight 1382 is by this airline?
The bureau of Transportation Statistics reports on-time performance for airlines at major u.S. airports. Jetblue, united, and uS Airways share terminal C at boston’s logan Airport. The percentage of on-time flights reported for August 2012 was 76.8% for Jetblue, 71.5% for united, and 82.2% for uS Airways ( bureau of Transportation Statistics website, october 2012). Assume that 30% of the flights arriving at terminal C are Jetblue flights, 32% are united flights, and 38% are uS Airways flights.ÂÂ
A. What is the most likely airline for Flight 1382? What is the probability that Flight 1382 is by this airline?
B. Suppose that an announcement is made saying that Flight 1382 will now be arriving late. What is the most likely airline for this flight? What is the probability that Flight 1382 is by this airline?
The Correct Answer and Explanation is :
This problem involves applying Bayes’ Theorem to calculate conditional probabilities. We are given the on-time performance of three airlines, their respective market shares, and a late flight event. Let’s break down the problem:
A. Most Likely Airline for Flight 1382 (On-time Flight)
We are given the following data:
- The probability that a flight is Jetblue, United, or US Airways:
- P(Jetblue) = 0.30
- P(United) = 0.32
- P(US Airways) = 0.38
- The on-time performance of each airline:
- P(On-time | Jetblue) = 0.768
- P(On-time | United) = 0.715
- P(On-time | US Airways) = 0.822
To find the most likely airline for Flight 1382, we need to use Bayes’ Theorem, which is formulated as:
[
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
]
Where:
- ( P(A|B) ) is the probability that event A occurs given event B.
- ( P(B|A) ) is the probability of event B given A.
- ( P(A) ) is the prior probability of event A.
- ( P(B) ) is the total probability of event B.
In this case, the “event B” is Flight 1382 being on time, and we want to calculate the probability that it is from each airline, which is:
[
P(Jetblue | \text{On-time}) = \frac{P(\text{On-time} | \text{Jetblue}) P(\text{Jetblue})}{P(\text{On-time})}
]
We calculate ( P(\text{On-time}) ) by considering all three airlines:
[
P(\text{On-time}) = P(\text{On-time} | \text{Jetblue}) P(\text{Jetblue}) + P(\text{On-time} | \text{United}) P(\text{United}) + P(\text{On-time} | \text{US Airways}) P(\text{US Airways})
]
B. Most Likely Airline for Flight 1382 (Late Flight)
For part B, we need to find the most likely airline given that Flight 1382 is late. We need to calculate the probability of each airline being the source of a late flight.
- P(Late | Jetblue) = 1 – P(On-time | Jetblue) = 1 – 0.768 = 0.232
- P(Late | United) = 1 – P(On-time | United) = 1 – 0.715 = 0.285
- P(Late | US Airways) = 1 – P(On-time | US Airways) = 1 – 0.822 = 0.178
Then, use Bayes’ Theorem again to calculate the conditional probabilities for each airline, given the event that the flight is late:
[
P(\text{Jetblue} | \text{Late}) = \frac{P(\text{Late} | \text{Jetblue}) P(\text{Jetblue})}{P(\text{Late})}
]
Where ( P(\text{Late}) ) is the total probability of a flight being late, which we calculate similarly to ( P(\text{On-time}) ), but with the late probabilities.
I’ll compute the exact values and generate a plot for this scenario now.
A. Most Likely Airline for Flight 1382 (On-time Flight)
After calculating the conditional probabilities using Bayes’ Theorem for an on-time flight:
- The probability that Flight 1382 is from Jetblue is 29.86%.
- The probability that Flight 1382 is from United is 29.65%.
- The probability that Flight 1382 is from US Airways is 40.48%.
Therefore, US Airways is the most likely airline for Flight 1382, with a probability of 40.48%.
B. Most Likely Airline for Flight 1382 (Late Flight)
For a late flight, we apply Bayes’ Theorem again using the late probabilities:
- The probability that Flight 1382 is from Jetblue is 30.47%.
- The probability that Flight 1382 is from United is 39.92%.
- The probability that Flight 1382 is from US Airways is 29.61%.
Thus, United is the most likely airline for Flight 1382 if it is reported late, with a probability of 39.92%.
Summary:
- Most likely airline for an on-time Flight 1382: US Airways (40.48% probability)
- Most likely airline for a late Flight 1382: United (39.92% probability)
I’ll now generate the image to help visualize this.
Here is the bar chart comparing the probabilities for Flight 1382 being from Jetblue, United, and US Airways, both for on-time and late arrivals. The chart should help visualize the differences in likelihood for both scenarios. Let me know if you’d like to explore further!
