statisticalrethinkin..

8.1. GOOD KING MARKOV AND HIS ISLAND KINGDOM 245(1) Wherever the King is, each week he decides between staying put for another weekor moving to one of the two adjacent islands. To decide his next move, he flips acoin.(2) If the coin turns up heads, the King considers moving to the adjacent island clockwisearound the archipelago. If the coin turns up tails, he considers instead movingcounter-clockwise. Call the island the coin nominates the proposal island.(3) Now, to see whether or not he moves to the proposal island, King Markov countsout a number of seashells equal to the relative population size of the proposal island.So for example, if the proposal island is number 9, then he counts out 9seashells. en he also counts out a number of stones equal to the relative populationof the current island. So for example, if the current island is number 10, thenKing Markov ends up holding 10 stones, in addition to the 9 seashells.(4) When there are more seashells than stones, King Markov always moves to the proposalisland. But if there are fewer shells than stones, he discards a number of stonesequal to the number of shells. So for example if there are 4 shells and 6 stones, heends up with 4 shells and 6 − 4 = 2 stones. en he places the shells and the remainingstones in a bag. He reaches in and randomly pulls out one object. If it is ashell, he moves to the proposal island. Otherwise, he stays put another week. As aresult, the probability that he moves is equal to the number of shells divided by theoriginal number of stones.is procedure may seem baroque and, honestly, a bit crazy. But it does work. e king willappear to move around the islands randomly, sometimes staying on one island for weeks,other times bouncing around without apparent pattern. But in the long run, this procedureguarantees that the king will be found on each island in proportion to its population size.You can prove this to yourself, by simulating King Markov’s journey. Here’s a short pieceof code to do this, storing the history of the king’s island positions in the vector positions:num.weeks