Skip to main content

Event types; airport weapon detection Bayes

Short answer

Mutually exclusive events cannot occur together (e.g., even and odd in a die toss). Exhaustive events cover the whole sample space (e.g., head or tail in a coin toss). Complementary events are both mutually exclusive and exhaustive (e.g., bulb working or not). (i) P(weapon) = 0.5×0.9+0.3×0.5+0.2×0.4 = 0.68. (ii) P(A|weapon) = 0.5×0.9/0.68 ≈ 0.662.

What do you mean by mutually exclusive, exhaustive and complementary events? Explain with examples. In a particular city, airport A handles 50% of all airlines traffic, airport B handles 30% and airport C handles 20%. The detection rates for weapons at the three airports are 0.9, 0.5 and 0.4 respectively. A passenger is randomly selected at one of the airports. Then (i) what is the probability that he/she carrying a weapon? (ii) If he/she is found to be carrying a weapon, what is the probability that airport A is being used?

Types of events

  • Mutually exclusive events: cannot occur together, so P(A ∩ B) = 0. Example: getting an even number and an odd number in one throw of a die.
  • Exhaustive events: together they cover the whole sample space, so at least one must occur. Example: getting a head or a tail in one toss. Events "even" and "odd" are exhaustive for a die.
  • Complementary events: A and its complement A′ are both mutually exclusive and exhaustive, with P(A) + P(A′) = 1. Example: a bulb is working (A) or not working (A′).

Airport problem

The detection rates are read as the probability that a weapon is detected (found) at each airport.

Let A, B, C be the events that the airport used is A, B, C, and W be the event that a weapon is found.

P(A) = 0.5   P(W|A) = 0.9
P(B) = 0.3   P(W|B) = 0.5
P(C) = 0.2   P(W|C) = 0.4

(i) Probability of a weapon being found (total probability)

P(W) = (0.5)(0.9) + (0.3)(0.5) + (0.2)(0.4)
     = 0.45 + 0.15 + 0.08
     = 0.68

(ii) Airport A, given a weapon is found (Bayes' theorem)

P(A|W) = P(A) P(W|A) / P(W) = 0.45 / 0.68 = 0.662

Answer: (i) 0.68, (ii) 0.662.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 1 questions →