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This question considers train connections between towns and a game of tag played on the train system.
The connections on the graph below represent train connections and walks through towns A, B, and C. Trains go from A→B,B→C,andC→A. A drunken man at any town will either walk around town or board a train to the next, each with probability 0.5.
- [2]
Write down the adjacency matrix for the directed graph.
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Find the total number of walks of length 7 from A to C.
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The drunken man has the same probability of traversing any sequence of 7 edges. Find this probability.
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Given that the drunken man starts at town A, find the probability that he will be at vertex C after traversing 7 edges.
Mika and Luca play tag across the three towns. Luca is ‘‘it.” Before starting the game, Mika cheated by attaching a tracker to Luca's backpack.
At each vertex, Luca has equal probability of advancing to the next vertex and staying to explore the town (and find Mika).
Mika, on the other hand, will advance to the next vertex if and only if Luca can move on the next turn and is on the vertex behind him (he will stay in the same town otherwise).
Trains leave towns at the same times.
Luca starts in town A, and Mika starts in town B.
The possible positions of the two can be represented by the following transition diagram. The first letter indicates the position of Luca, and the second indicates the position of Mika.
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Find the value of a.
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Find the probability that Mika is in town B and Luca is in town A after a large number of turns.
After a few turns, Luca believes that Mika is cheating. Luca moves forward and stays each with probability 21 and Mika tells Luca beforehand that he'll do the same. Luca does not know if Mika has started in town B or C.
The directed graph below represents the probabilities of advancing from state to state:
Find the value of
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a;
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b.
By grouping vertices "M before L" and "L before M," the graph may be reduced to the one below.
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Find c.
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Find the expected number of moves for Luca to catch Mika if Mika were not cheating.
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State the null and alternative hypotheses for Luca's significance test. Use the probability that Mika will be caught on any given turn.
- [3]
Find the smallest number of moves Luca can take before being 95% confident that Mika is cheating.
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