Question

The captain of a military ship has seen another ship in the distance. It is foggy and visibility is not great. You know the following facts. All ships in this area are either commercial ships or pirate ships: 90 percent are commercial ships and 10 percent are pirate ships. The captain identified the ship as pirate. The captain’s recognition capabilities have been tested in foggy conditions as part of his training. In his training, he made correct identifications 80 percent of the time and erred 20 percent of the time.

1. (a) What is the probability that a randomly selected ship in the area would be a pirate ship and that the captain would correctly identify it as such?

2. (b) What is the probability that a randomly selected ship in the area would be commercial and that the captain would incorrectly identify it as pirate?

3. (c) What is the probability that a randomly selected ship in the area would be identified by the captain as a pirate ship?

(d)What is the probability that the ship was a pirate ship, rather than a commercial one, given that the captain identified it as pirate?

Answer 1

P(C) = 0.9

P(P) = 0.1

The captain has identified the ship as a pirate ship.

P(Correct/P) = 0.8

P(Incorrect/P) = 0.2

P(Correct/C) = 0.8

P(Incorrect/C) = 0.2

(a) Pirate ship and correctly identified by the captain. (Here 'and' means intersection).

(b) Commercial ship and incorrectly identified by the captain as pirate

(c) Fundamental theorem of addition:-

P(E) = P(Correct/P) . P(P) + P(Incorrect/P) . P(P)

P(E) = 0.8 x 0.1 + 0.2 x 0.1

P(E) = 0.08 + 0.02

P(E) = 0.1

(d) Given that captain has identified the ship as a pirate ship, Probability that it is actually a pirate ship is 0.8