Question

A ship is carrying 30 travelers from various great houses on a long sea voyage from Braavos to King’s Landing where they will participate in the Game of Thrones:

• 5 are from House Targaryen(TAR), and 5 from House Lannister (LAN)
• 4 are from House Stark (STA) and 4 from House Tyrell (TYR)
• 3 are from House Baratheon (BAR) and 3 from House Martell (MAR)
• 1 from each of the following houses: House Arryn (ARR), House Tully (TUL), House Greyjoy (GRE), House Bolton (BOL), House Frey (FRE), House Mormont(MOR)

You are now asked to evaluate probabilities relating to this voyage.

Part 1 - Leaving the Ship

Only 9 of the 30 travelers will end leaving the ship when they reach King’s Landing. Based solely on the number of travelers per house, what is the expected value of

1. the number of travelers from LAN leaving the ship?
2. the number of travelers from STA leaving the ship?
3. the number of travelers from BOL leaving the ship?

Part 2 - Game of Thrones

The following 9 competitors end up leaving the ship and participating in the Game of Thrones:
3 from TAR, 2 from LAN, 2 from STA, 1 from GRE' and 1 from MAR.

At the end of the Game of Thrones,only 3 of the 9 competitors will win titles:

• 1 will win the Iron Throne (I),
• 1 will end up being the Hand of the King (H)
• 1 will be the King of the North (N)

Nobody can win more than one title.

1. In how many different ways can these 3 titles be distributed among the 9 competitors?
2. In how many different ways can these 3 titles be distributed among the 5 participating houses? (in other words we are asking you to figure out how many possible combinations of titles by houses there can be.) Your answer should not be derived by listing all the possibilities one by one. Instead you should derive your answer by reasoning with known formulas for permutations and combinations.
3. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that GRE will win at least one title?
4. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that LAN will win at least one title?
5. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that TAR will win at least one title?

Part 3 - Epidemic

Unfortunately for the competitors, Braavos had an epidemic of the Heineken virus. Grand Maesters work around the clock to learn as much as they can about the Heineken virus.

• In their findings, they concluded that 1.1% of people coming from Braavos will test positive for the Heineken virus.
• They also know that there is a false negative rate of 4% and a false positive rate of 0.6%.

Use the information above to calculate the probability that a traveler who tests positive does actually have the Heineken virus. For this exercise, define the following events:

• T+: testing positive
• T-: testing negative
• V: having the Heineken virus
• H: being healthy

Answer 1

Part 1 E( No a) of traveler from LAN leaving the ship) = x.pla) B El No. of travelou from STA leaving the ship) = .(x) g El No. of traveler from BOL leaving the ship) - 2.86%) Part 2 9 8 7 out those - of places a I we have to fill 3 compititors . ie we can in 9 x 8 x 7 waths or Ip places fill = 504

6 3 titles has to distributed among 5 houses je 5x4x3 ways or 50 - 60 - 9 - o title) atleast PLTAR will with title) = 1 PETAR will win 6x5x4 I 6P3 9x8x7 ( 9p3 P(LAN will win cetleat 1 title) - PULAN wilt will no (0) title) 7x8 x5 l 7p3) gx8x7 (9p3 12 - d) P(GRE will win cetteat & title) T I - PC GRE will win no tot tittet 9x8x7 Эр,

Part c lov HO (tt) + vetet I x (T-) - ve tat 10.04 1 1 0.006 y Since false positive = 0.6% false Negative a 4% Given RNM RAUL & Quort - P(Tt) z o.ot PTT-TREMEN TT-plekt -ooth = 0.989

Date Poor P/Aue/v) + Plave/v) = 1 Plaulu) a 1 - Pla-ve) = 1-0.04 = 0.96 ochila de