A little question on Probability

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Discussion Overview

The discussion revolves around the nature of probability, particularly focusing on events with probabilities of 0 (impossible events) and 1 (certain events). Participants explore examples from various contexts, including sports and natural phenomena, and question the philosophical implications of these probabilities. The conversation also touches on the concept of superhuman feats and the conditions under which certain events may be considered impossible.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that a probability of 0 indicates an impossible event, while others challenge this notion, suggesting that events can have a probability of 0 without being impossible.
  • A participant notes that in mathematics, certain events have defined probabilities, such as a die showing a number from 1 to 6 having a probability of 1, while showing a 7 has a probability of 0.
  • There is a discussion about the definition of "superhuman" feats, with some suggesting that if defined as something humans cannot do at all, the probability is 0, while others argue that if defined as something average humans cannot normally do, it gives a small probability.
  • One participant introduces the concept of the Boltzmann brain in relation to very small probabilities.
  • Another participant emphasizes that in continuous probability distributions, it is possible for specific outcomes to have a probability of 0, while still being part of a larger set of outcomes.
  • There is a suggestion that scoring 1000 points in a football game is not completely impossible, depending on certain conditions.
  • Participants express uncertainty about the implications of having events with probability 0 and the philosophical aspects of existence and certainty in relation to probability.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of probabilities of 0 and 1, with no consensus reached on the implications of these probabilities in real-world contexts. The discussion remains unresolved regarding the definitions and examples of impossible versus certain events.

Contextual Notes

Some statements reflect philosophical viewpoints on existence and certainty, which may influence interpretations of probability. The discussion also highlights the complexity of defining events in probability theory, particularly in continuous distributions.

Sundown444
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Well, I know that on a scale, a probability of 0 is impossible while a probability of 1 is certain. I do have some things down, but just in case, can anyone here please give me a list of what kinds of events are impossible (0) and what events are certain (1)? I would really appreciate it.

EDIT: And I don't mean card games or dice, but things like sports, cosmic events and natural phenomena like earthquakes and weather.

EDIT: Also, one more thing; What are the odds of a normal human doing some kind of feat that is superhuman in terms of superhuman strength, speed, agility and such? Is that impossible?
 
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For the real world, probabilities at the extreme edges depend a bit on your philosophical viewpoint. For example, we can all see your post, so are we absolutely certain it exists (as binary representation on a hard drive)? We could all just imagine it, without the post actually being there. Are we even sure we exist? But let's ignore those issues:
No probability for the real world is exactly 0 or 1 unless the event would necessarily violate a fundamental law of physics or is forbidden by logic, then its probability is 0.

In mathematics, it is easier: the probability that an (ideal) six-sided die gives a number from 1 to 6 is exactly 1. The probability that it gives 7 is exactly 0.

Sundown444 said:
What are the odds of a normal human doing some kind of feat that is superhuman in terms of superhuman strength, speed, agility and such? Is that impossible?
That depends on the definition of "superhuman". If you define it as "something humans cannot do at all", then the probability is 0 by definition, "something average humans cannot normally do" gives a small probability.
 
What makes you think that a probability of 0 is impossible?
 
Another relevant concept for very small probabilities is the Boltzmann brain.

Also note that every event "A happens" with a very small probability (or 0) has a corresponding event "A does not happen" with a very large probability (or 1).
 
Sundown444 said:
Well, I know that on a scale, a probability of 0 is impossible while a probability of 1 is certain.
Be careful! This is only true if you have a "discrete" probability. If your probability distribution is continuous, this is not true. For example, the probability of choosing any specific number between 0 and 1, with the uniform probability distribution, is 0, yet obviously some number has to be chosen!

I do have some things down, but just in case, can anyone here please give me a list of what kinds of events are impossible (0) and what events are certain (1)? I would really appreciate it.

EDIT: And I don't mean card games or dice, but things like sports, cosmic events and natural phenomena like earthquakes and weather.
How about scoring 1000 points in a football game?

EDIT: Also, one more thing; What are the odds of a normal human doing some kind of feat that is superhuman in terms of superhuman strength, speed, agility and such? Is that impossible?
 
HallsofIvy said:
How about scoring 1000 points in a football game?
Not completely impossible. You need a mad goalkeeper and probably a referee to extend the game too long (because 5.4 seconds per goal is a challenge), but it is certainly something that has a non-zero probability.
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h6ss said:
What makes you think that a probability of 0 is impossible?

Maybe because that is what I read when I looked up probability. Now that you have said that, is there something I am missing?
 
Sundown444 said:
Maybe because that is what I read when I looked up probability. Now that you have said that, is there something I am missing?

Only events that are impossible have zero probability, but the converse is not true in general. In fact, it's possible for an event to have probability zero without being the empty event. For example, the event of flipping only tails infinitely many times in a row. This event has probability zero, but it is not empty, because the underlying probability space is the set of all possible ways to flip a coin infinitely many times. What would be impossible, however, is flipping banana, since the coin has only a heads side and a tails side, and no sides marked with the word banana.
 
Sundown444 said:
Well, I know that on a scale, a probability of 0 is impossible while a probability of 1 is certain.

What you "probably" mean is that "an event with probability zero is impossible" - not that "a probability of zero is impossible".

The formal theory of mathematical probability deals with probability spaces and the probability of events in those spaces. It doesn't deal with whether those events actually happen or not. So questions about the possibility or impossibility of events in a probability space involve opinions about applying mathematical probability to situations. They aren't questions that are answered by the axioms of probability theory.
 

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