I have a question concerning what we mean by probability.
I've thought about this a fair bit and havn't come up with a good
answer.
If the probability of somthing happening is 1/3 this means that if we
chance it 3 time the most likely outcome is that it happens once.
After 3 goes
P0 8/27 (probability of 0 occurances)
P1 12/27 ( " 1 " )
P2 6/27
P3 1/27
but we havn't defined probability, we're still using the concept in
our 'definition'.
It would seem that probability tells us nothing about the results of an experiment if we
do it finite times, ok so the probability of a positive outcome is 1/3, but
you could do it 1000000 times and still not get a positive, or you could
get all positive.
The only way we can really define it (it seems to me) is to do
the experiment infinite times then we can say that a third of the time
we would get a positve result, but even that's no kind of definition, firstly, because we can't do it infinite times. And secondly , and possibly more importantly, because I KNOW that probability does mean something when we're only doing an experiment a finite number of times (Hell I've been to Vegas).
What you’re struggling with, murphyboy99 is not probability per se, but randomness, a property that probability is based on. This is one of the reasons people talk about a “fair” die, or a well-shuffled deck of cards. It’s because there has to be some level whereby the outcome of the test is completely unknowable. This “randomness” is a property that is well-studied by those in computer science as random number generation can be a tricky subject if you aren’t careful.
When you get into actual events, you’re dealing with (generally) Bayseian statistics that model the likelihood of a string of events to occur. As you correctly noted, there is a non-zero probability that the MOST unlikely string of events will occur. This is a peculiar result of randomness.
One of the ways we have historically defined “order” and “randomness” is through the ability for us to determine patterns. To that end, if a random number generator exhibits a patterned output, we are likely to say that the patterned output indicates the process whereby the creation of this pattern occurred was not random. This, however, is a spurious conclusion. Since we now have mathematical models available for the generation of truly random tests, we have more of a mechanistic view of randomness today than we did before. But this more rigorous approach to randomness demands we delve deeply into chaos theory and the like. I recommend James Gleick’s book, Chaos: Making a New Science.
Historically, there have been many different interpretations of what it means to say “the probability of an event E is p”. Many brilliant philosophers have failed to resolve this question, so you’re not likely to get a conclusive answer here. I’ll see if I can’t dig up a source, but the discussion tends to get very technical, so there may be nothing forthcoming.
You don’t have to actually do an infinite number of trials, you can take a limit instead. Suppose, for instance, I have a fair, six-sided die. What does it mean to say that the probability of rolling a 6 is 1/6? It means that as I make more and more rolls, the number of sixes that come up will approach arbitrarily close to one sixth of the total number of rolls. Yes, maybe after the first hundred rolls, I haven’t gotten a single six. Highly unlikely, but it might happen. Then I just keep on rolling. After 10,000 rolls, the result of those first hundred will only influence the final answer by a percent.
There is still the question of how many rolls of the dice can be considered “enough”, but to answer that, you need to step a little bit past just probability, to statistics.
It seems to me that most of what we call probability is really based on the extent of out knowledge of the situation. Before a surgery, you could say that the probability of a hemorrhage is 1 in 10,000, but if you knew that the patient had an aneurysm near the site, you’d raise the odds. Without that knowledge, your odds take the possibility into account.
Like rolling dice. When they’re tossed, they are “pre”-destined to land with certain numbers face up, we just don’t know the variables of the trajectories, rotations, etc to do the math.
Unless you base your randomness on quantum phenomena, “probability” is just our approximation of the situation based on our limited understanding. In any situation, it really is pre-ordained- we just don’t know ahead of time what it is.
JS Princeton. I’m pretty sure that it is probability that I’m having problems defining, and not randomness.
Chronos, Here in lies the rub, when I’ve thrown the die 100 times I could either get 0 sixes or 100 sixes. When I throw the dice 100,000,000 times again I might get 0 sixes or 100,000,000 sixes. I know that you say it’s highly unlikely and I agree (The odds against it are so small my desktop calculator won’t work them out). The point is I can’t tell you anything about the outcome of the experiment without using the term odds, likelihood or probability. But what I’m after is a thought experiment that will allow me to define the term probability; this one doesn’t cut it. The only way I can think of doing it is by allowing the experiment to run to infinite goes, then I could say that exactly 1/6 of the outcomes would be sixes.
My Physics professor once told me that ‘probability tells us nothing about the outcome of a single event’. If that’s true does it tell us anything about the outcome of any non-infinite set of events ?
It’s strange to me that we all have an intuitive understanding of probability, and can use it in real life, but can’t explain it without using infinity, which is not intuitive and not used in everyday life.
Ultrafilter, I’m beginning to agree, I mean if I can’t get a handle on this, what chance did the most brilliant philosophers through history ever really have ?
Then perhaps my definition is naive, but I would say that probability is the ratio of, given specified knowledge about a situation, the number of ways a specific outcome can occur divided by the number of all possible outcomes. As other posters have said, this ratio does not help you to predict a specific event or even a lot of identical events. We can only say that the ratio of actual occurrences to actual trials has a tendency to approach the probability.
Disclaimer: Had two semesters of statistics in college (i.e., long time ago), IANAM.
Not quite sure what you mean, but probability is very well-founded, in terms of measure theory. This is, of course, only one interpretation of probability, but those others that don’t have rigorous theoretical foundations are non-mathematical.
