Answer :
The probability that Taylor will be late for her date in Vegas is 0.
To determine the probability that Taylor will be late, we need to calculate the total time it will take for her to take off and compare it with the time left before her important date.
The flight time is exactly 13 hours, and we are given that there are two airplanes ahead of Taylor's plane, with each plane taking off every 10 minutes on average. Since Taylor's plane is third in line, we can calculate the time until her plane can take off as follows:
Time for two planes to take off = 2 planes × 10 minutes/plane = 20 minutes
We need to convert this time into hours to compare it with the total time available:
20 minutes ÷ 60 minutes/hour = 1/3 hour ≈ 0.333 hours
Now, let's add the flight time and the time Taylor has to wait before taking off to find the total time from now until she lands in Vegas:
Total time = Flight time + Wait time
Total time = 13 hours + 0.333 hours
Total time ≈ 13.333 hours
Taylor has 13.5 hours until her date, so the total time required for Taylor to fly to Vegas and be on time for her date is less than the time available. Therefore, we can calculate the probability of Taylor being late as follows:
Total time required < Time until date
13.333 hours < 13.5 hours
Since the total time required for Taylor to arrive in Vegas is less than the time until her date, the probability that she will be late is 0. Taylor will be able to make it to her date on time, assuming there are no other delays.
The probability of her flight departing immediately or being delayed due to other planes, we arrived at the probability of [tex]\( \frac{26}{27} \)[/tex]. This means there is a very high likelihood that Taylor will be late for her date
To calculate the probability that Taylor will be late for her date, we need to consider the time it takes for her flight to depart and the time it takes for her to arrive at Vegas.
1. Flight Departure Time:
Taylor's flight departs 13 hours before her date.
2. Time to Get to Vegas:
Her flight takes 13.5 hours, so she will arrive 0.5 hours before her date.
3. Probability of Delay:
Given that there are three planes in line for takeoff and one plane takes off every 10 minutes, the probability that Taylor's plane will take off immediately is 1/3. If it doesn't take off immediately, it will take approximately 10 minutes per plane ahead of her for her plane to take off.
The probability of her being late is the probability of her plane not taking off immediately plus the probability of delays due to other planes:
[tex]\[ P(\text{late}) = \frac{2}{3} + \left( \frac{1}{3} \times \frac{2}{3} \right) + \left( \frac{1}{3} \times \frac{1}{3} \times \frac{2}{3} \right) \][/tex]
[tex]\[ P(\text{late}) = \frac{2}{3} + \frac{2}{9} + \frac{2}{27} \][/tex]
[tex]\[ P(\text{late}) = \frac{18}{27} + \frac{6}{27} + \frac{2}{27} \][/tex]
[tex]\[ P(\text{late}) = \frac{26}{27} \][/tex]
So, the probability that Taylor will be late for her date is [tex]\( \frac{26}{27} \).[/tex]