Brilliant food
You, your host, and 18 other guests (a total of 20 people) are sitting around a round table. You are seated right across the host. In other words, if the seats are numbered starting with the host's seat (#1), you are sitting on seat #11.
The host has a dish with 19 portions of food. She takes one, and then passes the dish to one of her neighbors (with equal probability). Similarly, this guest will take a portion, and pass the dish with equal probability to one of his two neighbors (either back to the host or to the guest next to him). If the person who gets the dish already has a portion of food, that person will not take more, but just pass the dish to one of his/her two neighbors (randomly). This general rule applies at each step until all the food is served.
What is the probability of you ending up hungry (i.e. you being the last one receiving the dish)?
Give the answer in percentage, to two decimal places.
The answer is 5.26.
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1 solution
For you to be the last, two conditions have to be satisfied. #1. First the serving dish has to get to one of your neighbors. #2. Second, it has to go around the table and reach your other neighbor. There is no other way for you to be the last.
The probabilities of passing the dish do not depend on the amount of food on the serving dish or on the food in front of the person passing the dish. Therefore the history of passing (the path the serving dish took) before condition #1 is satisfied is irrelevant.
If that is true, we can re-state the question this way: What is the probability of you being last, assuming that the dish starts next to you, but some of the guests has already taken food?
Notice that the number of guests taking food from the dish before it reaches your first neighbor does not matter. The serving dish has to go around the table to your other neighbor. If someone is already served, that person is not taking more food. By the time the serving dish reaches your other neighbor all food, except the one portion he will take, will be gone. Again, this does not depend on the path the dish takes.
In this new approach, it is evident that your position around the table is irrelevant: you may be sitting next to the host, opposite to her or anywhere else, the probability is the same. Each one of the guests has the same probability of being last. With 19 guests the probability is 1/19=5.26%.