A subset of King Robert I’s small council has assembled, per instructions from t

Question

A subset of King Robert I’s small council has assembled, per instructions from their king, for a secret conclave. In attendance are:

Jon Arryn - Hand of the King

Pycelle - Grand Maester

Petyr Baelish - Master of Coin

Renly Baratheon - Master of Laws

Stannis Baratheon - Master of Ships

Barristan Selmy - Lord Commander of the Kingsguard

The six members enter a room and take seats around a round table. Before their meeting, however, Varys (Master of Whisperers) mounted a tiny device that can shoot a poisoned dart underneath the table at one of the six seats. He was not entirely sure where each individual would sit at this conclave, but his network of spies has gathered a great deal of information about the habits and motives of all the attendees, and with that information Varys has come up with the following calculations:

i. When the individuals enter the room, there is a 44% chance that Renly Baratheon will choose the seat where the device is mounted, a 42% chance that Petyr Baelish will choose that seat, and a 14% chance that Pycelle will choose the seat with the poisoned dart device.

ii. If the device is mounted at Renly’s seat, there is a 34% chance that he’ll notice it. If he does notice it, there’s a 31% chance he’ll use it to poison Petyr Baelish, a 16% chance he’ll use it to poison Jon Arryn, a 1% chance he’ll use it to poison his brother, Stannis, and a 52% chance that he won’t use it at all.

iii. If the device is mounted at Petyr Baelish’s seat, there is a 99% chance that he’ll notice it. If he does notice it, there’s a 20% chance he’ll use it to poison Jon Arryn, a 36% chance he’ll use it to poison Pycelle, a 22% chance he’ll poison Renly, a 15% chance he’ll poison Stannis, a 2% chance he’ll poison Barristan, and a 5% chance he won’t use it to poison anyone at all.

iv. If the device is mounted where Pycelle is sitting, there is an 86% chance that he’ll notice it. If he does notice it, there’s a 10% chance he’ll use it to poison Petyr Baelish, a 10% chance he’ll use it to poison Jon Arryn, and an 80% he won’t use it at all.

v. (If someone does not notice that the device is mounted at his seat, then he won’t use it to poison anyone. The device is obscure enough that no one would accidentally set it off.)

Assuming Varys’s calculations are correct, if Jon Arryn is poisoned by the dart, what is the probability that the dart was mounted at Renly’s seat?

Explanation / Answer

It is a case of conditional probability.

We need to find out:

P(Dart at Renly's seat | Jon Arryn was poisoned) = P( Jon Arryn was poisoned | Dart at Renly's seat)*P(Dart at Renly's seat) / Total probability of Jon Arryn was poisoned

P( Jon Arryn was poisoned | Dart at Renly's seat) = 0.16*0.34

P(Dart at Renly's seat) = 0.44

P( Jon Arryn was poisoned | Dart at Petyr's seat) = 0.20*0.99

P(Dart at Petyr's seat) = 0.42

P( Jon Arryn was poisoned | Dart at Pycelle's seat) = 0.1*0.86

P(Dart at Pycelle's seat) = 0.14

P(Dart at Renly's seat | Jon Arryn was poisoned) = (0.16*0.34*0.44) / ((0.16*0.34*0.44) + (0.20*0.99*0.42) + (0.1*0.86*0.14))

= 0.2009

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