Upper and lower probabilities

Imagine you have a bag of marbles. You don't know how many marbles are in the bag, but you want to guess. You ask people around you for their guesses and you get a different answer from each person.

But you notice that some people's guesses are more likely to be correct than others. For example, if one person is known to be really good at guessing how many marbles are in a bag, then you might trust their answer more than someone who is known to be bad at guessing.

This is kind of like upper and lower probabilities. Upper probability is like the "best case scenario" for something happening. It's the highest chance that the event could occur, based on the available information. Lower probability is like the "worst case scenario." It's the lowest chance of the event occurring.

If you go back to the bag of marbles example, the upper probability for how many marbles are in the bag might be the highest guess you heard from someone you trust. The lower probability might be the lowest guess you heard from someone not so trustworthy.

So when people talk about upper and lower probabilities, they're basically saying what the best and worst possible chances are for that event happening, given what they know.