Question

1. Once torists arrive in their hotel in Dublin, they mainly want to enjoy shopping on Grafton Street and a day at Dublin Zoo the next day. From survey data, when tourists check in to their hotel in Dublin, 60% of tourists plan to go to Grafton Street the next day, 30% to Dublin Zoo, and only 10% stay at their hotel, when on Grafton Street, 50% return to Grafton Street the next day, 30% plan to go to Dublin Zoo the next day, and 20% stay in the hotel. Finally, when at Dublin Zoo 30% return the next day, 30% go Grafton Street the next day, while 40% stay in their hotel the next day (a) Construct a transition diagram, clearly labelling the probabilities, to represent this situation. [5 marks] 5 marks (b) Construct a transition matrix to represent this situation. (c) Starting from the hotel, what is the probability (in percentage) that a tourist will spend their final day in the hotel at the end of a 4-day trip. [8 marks] (d) Determine the long-term probability that a tourist spends the final day of their [14 marks]

Answer 1

Solution Giiven that once toursts arrile tn their hotel in Dublin, 4hey mainty coant to enjoy shopping on 6roon Steet and a day ot Dublin zoo Let the States of the Martov chain are Meaning Stotes at Dublin zoo at hotel tn Dublin on Gton strect To dtoga 0.3 6 0.3 b Transttion matrix Current stałe GLo.3 o2 0-5

2 Let TT n) beirg at these plaes on nth day of Starting fom the hotel to) holel (o Potobilty distibution on ihda (o 03 0.4 0-3 5 3 0.2 0.5 0.3 .262 .4638 o3 0365 04635 0.3 2364 oVK36 0.3 0.2365 0.4635 Theretore 4) TH0 2365 the Probabtlity at a tourist coill send thus -fral day in 4he hotel on ith day is 02365

d, long-term probab. lity tAtibution disfies TP. T 63 0.10-6 03 02 0.5 Myニ0.YTD to.ITTH+0.2Tg O-子 TTH 26 26 Tu - 26 io ib 0-3 Requied tobabili 0.2