Question

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. 
Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55\%, 37\% and 17\% due to severe, moderate and light turbulence respectively. 
On the basis of the above information, answer the following questions: 
Find the probability that an airplane reached its destination late. If the airplane reached its destination late, find the probability that it was due to moderate turbulence.

Hint:

Use the law of total probability to calculate overall probabilities and Bayes' theorem for conditional probabilities when given dependent conditions.

Answer 1

Step 1: Given Information. Turbulence can be severe, moderate, or light, each occurring with equal probabilities: \[ P(\text{Severe}) = P(\text{Moderate}) = P(\text{Light}) = \frac{1}{3}. \] The probability of an airplane reaching late due to: \[ P(\text{Late}|\text{Severe}) = 0.55, \quad P(\text{Late}|\text{Moderate}) = 0.37, \quad P(\text{Late}|\text{Light}) = 0.17. \].
Step 2: Find the probability that an airplane reached its destination late. Using the law of total probability: \[ P(\text{Late}) = P(\text{Late}|\text{Severe})P(\text{Severe}) + P(\text{Late}|\text{Moderate})P(\text{Moderate}) + P(\text{Late}|\text{Light})P(\text{Light}) \] Substituting the values: \[ P(\text{Late}) = (0.55 \cdot \frac{1}{3}) + (0.37 \cdot \frac{1}{3}) + (0.17 \cdot \frac{1}{3}) \] Simplifying: \[ P(\text{Late}) = \frac{0.55 + 0.37 + 0.17}{3} = \frac{1.09}{3} = 0.3633. \]
Step 3: Find the probability that it was due to moderate turbulence. Using Bayes' theorem: \[ P(\text{Moderate}|\text{Late}) = \frac{P(\text{Late}|\text{Moderate})P(\text{Moderate})}{P(\text{Late})} \] Substituting the values: \[ P(\text{Moderate}|\text{Late}) = \frac{(0.37 \cdot \frac{1}{3})}{0.3633} \] \[ P(\text{Moderate}|\text{Late}) = \frac{0.37}{3 \cdot 0.3633} = \frac{0.37}{1.09} = 0.3394. \] 
Final Answers: The probability that an airplane reached its destination late is: \[ P(\text{Late}) = 0.3633. \] The probability that the airplane was late due to moderate turbulence is: \[ P(\text{Moderate}|\text{Late}) = 0.3394. \]