Question

Airplanes are by far the safest mode of transportation when the number of transported passengers is measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is \( 0.00001\% \). Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let \( E_1 \) be the event that there is a plane crash and \( E_2 \) be the event that there is no crash. Let \( A \) be the event that passengers survive after the journey. On the basis of the above information, answer the following questions:
(i) Find the probability that the airplane will not crash.
(ii) Find \( P(A \,|\, E_1) + P(A \,|\, E_2) \).
(iii)
(a) Find \( P(A) \).
OR
(b) Find \( P(E_2 \,|\, A) \).

Hint:

When dealing with probabilities of complementary events, remember that the sum of their probabilities is always 1. If \( P(E_1) \) is the probability of an event, then \( P(E_2) = 1 - P(E_1) \).

Answer 1

(i) Probability that the airplane will not crash: 
The probability of a plane crash is: \[ P(E_1) = 0.00001\% = \frac{0.00001}{100} = 10^{-7}. \] The probability that the airplane will not crash is: \[ P(E_2) = 1 - P(E_1) = 1 - 10^{-7}. \] 
Solution: 
\[ \boxed{P(E_2) = 1 - 10^{-7}.} \]