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I have a question about what types of situations we are able to assign a probability. Let me make up a hypothetical scenario here that roughly gets at what I'm trying to understand.

MY PREMISE: Suppose we know that there exists 1 billion red marbles in the world, and furthermore we know that every one of these marbles was invented and produced by Bob. These are the only red marbles that we are aware exist. Now suppose that we find a new red marble.

MY QUESTIONS: What is the probability that this new red marble that we have discovered, was made by Bob? And how do you come up with that calculation? Is it even possible to assign a probability to this situation?

Now, if we can't figure out a probability to this situation, I have a few follow up questions:

A) Why can't we assign a probability to this situation?

B) Can we even say that it's more likely that the red marble was made by Bob, than not made by Bob? My intuition about this question says yes, it does seem reasonable to say that it is more likely that the red marble was made by Bob, than unlikely. But I'm also aware that intuition is often wrong in cases like this, which is why I'm asking what the mathematics and logic says about this.

C) What if we knew that Bob had made 5 billion red marbles (as opposed to 1 billion)? Is it more likely now (compared to Bob only having made 1 billion) that the new red marble was made by Bob?

By the way, feel free to change my hypothetical scenario or use a different example if it helps to get at the heart of the question I'm trying to understand here. And thanks in advance for any insights you can provide.

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To assign a definite probability, you must have a thorough understanding of what you currently know— your model of the world.

To pick a different example, suppose you flip a coin and it lands on the table. What is the probability that it comes up heads?

  • A person with no knowledge of the coin is forced by symmetry to conclude that it is just as likely to come up heads as tails.
  • A person who watches a slow-motion replay of the coin as it falls, but the video cuts out before the coin lands, may be able to assign a more definite probability.
  • If I take a look at the coin and see where it's landed, I may be able to assign even 100% probability that it lands one way or another—because now I know.

All three of these probability assignments are simultaneously correct, because all three of them are true to the person who is assigning values based on the precise state of their present knowledge.

Probabilities must be based on thorough understanding of present knowledge. If it's not clear what you know (e.g. about how marbles are made, or about how many people may be making marbles, etc.), then the problem can be under-defined: you can't assign a number because you're missing important information.

To fix the marble scenario, you might need to make some of your assumptions explicit: marbles are made by a certain process, only humans make marbles, one person has a monopoly on this process, I know of very few ways that someone else could make marbles in this world, etc. These assumptions are what guide your intuition that probabilities should go certain ways.

Some people relate probability assignments to betting: in this red-marble world, how much would you be willing to bet that the marble was made by Bob? If you wouldn't be willing to bet all of your money, what reasons might you have in mind? Knowing those reasons may help you assign proper probabilities.

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  • $\begingroup$ Thanks for the response. I'm going to stew over what you wrote and I'll probably come back with more questions. $\endgroup$ – WA1NGRO Mar 20 '18 at 0:46
  • $\begingroup$ @WA1NGRO I recommend Edwin Jaynes' "Probability Theory: The Logic of Science". While there are some issue with it (it was incomplete, the Cox derivation doesn't actually work), Jaynes does a good job explaining probability conceptually and how to appropriately apply it. For example, there are commonly used theorems that have very unrealistic assumptions. Many issues in applying statistics are due to incorrectly formulating the problem, not the math. Many people worry if their data are "really" normally distributed but this misunderstands the normal distribution. $\endgroup$ – Derek Elkins Mar 20 '18 at 2:46

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