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Your road trip is not going so well, a traffic jam delayed your schedule and you had to stop in a city you don't know to find a hotel. In a quick search on the internet, you found 3 hotels and decided that you will give them a shot. You may visit any number of hotels but once you leave a hotel you will not go back to back to it, you are just too tired for the trip back and forth. If you proceed optimally, what is the chance that you choose the best hotel to stay in?
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1 solution
Let's number the hotels according to their quality being 1 the best hotel and 3 the worst. Let's choose a hotel at random to be the first one we visit and use it as a baseline, then we pick the first hotel which is better than the first one, this will work if we visit in the orders:
(2, 1, 3), (2, 3, 1) and (3, 1, 2)
and will fail if we visit the hotels on the orders:
(1, 2, 3), (1, 3, 2) and (3, 2, 1)
this gives you a chance of 1/2 of choosing the right one which is better than the 1/3 chance if you choose at random and it turns out it is the optimal solution in this particular case of the optimal stopping problem