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: 
(i) Find the probability that an airplane reached its destination 
(ii)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

Information Provided:
1. Turbulence severity (Severe, Moderate, Light) occurs with equal probability of 1/3 each:  

\(P(\text{Severe}) = P(\text{Moderate}) = P(\text{Light}) = \frac{1}{3}.\)

2. Conditional probabilities of an airplane being late given turbulence type:  

- Severe turbulence:  \(P(\text{Late}|\text{Severe}) = 0.55\) 
- Moderate turbulence:  \(P(\text{Late}|\text{Moderate}) = 0.37\)
- Light turbulence:  \(P(\text{Late}|\text{Light}) = 0.17\). 

(i) Calculate the overall probability of an airplane arriving late. Employing 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 given values:

\(P(\text{Late}) = (0.55 \cdot \frac{1}{3}) + (0.37 \cdot \frac{1}{3}) + (0.17 \cdot \frac{1}{3}).\)

Simplification yields:

\(P(\text{Late}) = \frac{0.55 + 0.37 + 0.17}{3} = \frac{1.09}{3}.\)

Therefore:

\(P(\text{Late}) \approx 0.3633.\)

(ii) Given that an airplane arrived late, determine the probability that this lateness was due to moderate turbulence.  

Applying Bayes' theorem:

\(P(\text{Moderate}|\text{Late}) = \frac{P(\text{Late}|\text{Moderate})P(\text{Moderate})}{P(\text{Late})}.\)

Substituting values:

\(P(\text{Moderate}|\text{Late}) = \frac{(0.37 \cdot \frac{1}{3})}{0.3633}.\)

Further simplification:

\(P(\text{Moderate}|\text{Late}) = \frac{0.37}{3 \cdot 0.3633} = \frac{0.37}{1.09}.\)

Consequently:

\(P(\text{Moderate}|\text{Late}) \approx 0.3394.\)

Final Results: 1. The probability of an airplane arriving late is approximately 0.3633.

\(P(\text{Late}) \approx 0.3633.\)

 2. The probability that the airplane's lateness was caused by moderate turbulence is approximately 0.3394.

\(P(\text{Moderate}|\text{Late}) \approx 0.3394.\)