Question

4. Suppose you are the Lord Commander of the Night’s Watch. You have 3 identical armies and 3 different castles (Castle Black (1), Eastwatch (2), and the Shadow Tower (3)) to defend. A possible defense arrangement is a placement of armies at the 3 castles. Castles can be left without an army to defend it. Recall that using the stars and bars counting technique, there are 3+2 2 = 5 2 different defense arrangement

(c) (9 points) Recall that each defense arrangement corresponds to a binary string of length 5 with 2 ones. Consider this alternative random strategy: Construct a length 5 binary string one bit at a time by flipping a coin. If the coin comes out heads, put a 1. If the coin comes out tails, put a 0. If you ever have 2 ones stop and fill the rest of the positions with zeros. If you ever have 3 zeros, fill the rest of the positions with ones. i. What is the probability that you put all armies on Castle Black? ii. What is the probability that you put one army each on Castle Black, Eastwatch, and the Shadow Tower? iii. Will this strategy result in a uniform distribution over all possible defense arrangements? Why or why not?

(d) (6 points) Design your own strategy that will result in a uniform distribution over all possible defense arrangements.

(c) (9 points) Recall that each defense arrangement corresponds to a binary string of length 5 with 2 ones. Consider this alternative random strategy: Construct a length 5 binary string one bit at a time by flipping a coin. If the coin comes out heads, put a 1. If the coin comes out tails, put a 0. If you ever have 2 ones stop and fill the rest of the positions with zeros. If you ever have 3 zeros, fill the rest of the positions with ones. i. What is the probability that you put all armies on Castle Black? ii. What is the probability that you put one army each on Castle Black, Eastwatch, and the Shadow Tower? iii. Will this strategy result in a uniform distribution over all possible defense arrange- ments? Why or why not? (d) (6 points) Design your own strategy that will result in a uniform distribution over all possible defense arrangements

Answer 1

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