Thompson Scattering is presented in the AI Crash course book as a method to select from multiple candidates offering repeated trials. The canonical example is the "one-arm bandit" problem-- trying to select the best slot machine or the best marketing strategy from several options.
Suppose you had 5 slot machines, and you wanted to determine which provides the best "return". A "default"/naive approach to determine which slot machine provides the return is to run a certain number of trials (say, 2000) and see how much money is returned. The one returning the most money is presumably the best. After this experiment you will have invested $10000 and probably gotten $1500 back. Thompson scattering is a more adaptive techinque- after each trial you keep a list of how many times you put money in and either won or lost. With each trial, you look at the odds of winning from each machine you have accrued to that point and make a new random decision of which machine to pick, using a Beta function to give the machines which have performed best TO THAT POINT better odds. You will still lose money, but you will likely lose less.
Thompson scattering seems ideal for events with repeated short trials where one item must be selected from a group.
Other applications of Thompson scattering might include:

• Picking restaurants. A Thompson Scattering approach would involve trying each of the 5 restaurants and then finding which ones made you wait longer than others, picking a "failure" if you waited more than 10 minutes or a "success" if you waited less. You could also apply this to different periods of time at the same restaurant (i.e. 11:30, 11:45 or 12:00 at Wendy's vs similar times at Burger King).
• Picking the best marketing