Klingons on the starboard bow?
created by DALL·EThe puzzle Klingons on the starboard bow? is
“Captain, there is an alien ship approaching on the starboard bow,” said Chekhov.
“What kind of vessel is it?” enquired James T. Kirk, commander of the Starship Enterprise.
“A Klingon vessel, captain, I think,” said the ensign.
“Analysis, Mr Spock,” said the commander to his first officer.
“It could well be a Klingon vessel, captain, 90 per cent of ships in this sector are Klingon.”
“And the other 10 per cent are Romulan vessels?”
“Precisely, captain.”
“Well, Spock, as you know, we are at war with the Klingons and have recently signed a peace treaty with the Romulans. I am wary of firing a photon torpedo after the debacle of the M5 computer.* Starfleet are insisting that we be 95 per cent confident a ship is of Klingon origin before we fire on it. How good is ensign Chekhov at identifying Klingon vessels?”
“His records show that the probability he says it’s a Klingon ship when it is a Klingon ship is 98 per cent and …”
“That’s all the evidence we need,” interrupted the captain. “Fire the photon torpedoes.”
Has Captain Kirk made the right decision or should Spock step in and voice his concern?
Solution
To determine if Captain Kirk made the right decision we need to calculate the probability that the ship is indeed Klingon, given Chekhov’s identification capabilities.
Let’s denote this probability, \(\text{P(K given C)}\): the probability that the ship is Klingon (\(\text{K}\)) given Chekhov says it is Klingon (\(\text{C}\)).
From the dialogue we know:
- \(\text{P(K)=0.9}\): 90% of ships in this sector are Klingon.
- \(\text{P(R)=0.1}\): 10% of ships in this sector are Romulan.
- \(\text{P(C given K)=0.98}\): If the ship is Klingon, Chekhov correctly identifies it as Klingon 98% of the time.
We apply Bayes’ theorem:
\[\text{P(K given C)} = \frac{\text{P(C given K)⋅P(K)}}{\text{P(C given R)⋅P(R) + P(C given K)⋅P(K)}}\]
Bayes’ theorem gives the probability of an event (here \(\text{P(K given C))}\) based on new information that is related to that event. In our case, we know that 10% of ships in this sector are Klingon (\(\text{P(K)}\)) and we want to know how this is changed/updated based on Chekhov’s performance in identifying Klingon ships, \(\text{P(C given K)}\).
The only unknown is on the right-hand side, \(\text{P(C given R)}\). This is the probability Chekhov misidentifies a Romulan ship as Klingon.
Without this information, we cannot calculate the exact probability. However, we can determine the maximum value of \(\text{P(C given R)}\) that would still meet Starfleet’s 95% confidence requirement.
That is, we want \(\text{P(K given C) > 0.95}\).
Solving the inequality we end up with \(\text{P(C given R) < 0.46}\).
This means that if Chekhov misidentifies Romulan ships as Klingon less than 46% of the time, the confidence level would be above 95%, meeting Starfleet’s requirement.
Given Chekhov’s high accuracy with Klingon ships (98%), it’s unlikely his error rate for Romulan ships would be as high as 46%. In fact, this is worse than a random guess! Therefore, it’s probable that the actual confidence level is above 95%.
Captain Kirk is right to fire the torpedoes because he knows that Chekhov would be able to recognise Romulan ships at least 46% of the time!
