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 arrangements. (a) (4 points) List out all possible defense arrangements. (b) (9 points) Now, the White Walkers will be coming to attack your castles. You do not know which castles they will come to so you want to distribute your armies randomly. Your random strategy is as follows: Roll a fair six-sided dice three times. After each roll, if you roll a 1 or a 2, put an army on Castle Black. If you roll a 3 or a 4, put an army on the Eastwatch. If you roll a 5 or a 6 put an army on the Shadow Tower. i. What is the probability that you put all of your 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?

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 ossible 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 (2)) different defense arrangements 3+2 (a) (4 points) List out all possible defense arrangements. (b) (9 points) Now, the White Walkers will be coming to attack your castles. You do not know which castles they will come to so you want to distribute your armies randomly. Your random strategy is as follows: Roll a fair six-sided dice three times After each roll, if you roll a 1 or a 2, put an army on Castle Black. If you roll a 3 or a 4, put an army on the Eastwatch. If you roll a 5 or a 6 put an army on the Shadow Tower i. What is the probability that you put all of your 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!

Answer 1

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