Since whenever I hear people say the odds are “30 000 to 1” on something, I always wonder exactly how they find it out or know what the exact number or range is correct.
What is the math behind probability theory?
How are statistics gathered?
The answer to either of those questions could fill several really big textbooks, so it’s kinda tough to distill into one post. You can find an introduction to the math behind probability theory here.
Like what Ultrafilter said, but I can give you two really dumbed down quickie answers to your two questions.
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You look at the total number of possibilities and the relationship of that total to the event you are investigating. For example, a die has 6 sides, so the chance of it coming up 1 on one throw is 1 in 6, or 1/6, or 16.67%
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Experiments, polls, and pure mathematical reasoning.
Just to clear up a point of notation, as you used “20,000 to 1”: I think “1 in 6” is the same as “5 to 1”. The former being more common in maths, the latter in conversation.
Basic Probability:
When you have a finite number of equally likely outcomes to an event, you can place an exact mathematical probability on each outcome (i.e. the roll of dice).
Basic Statistics:
Many real-world events don’t have such a nice cleanly defined outcome though, like predicting tomorrow’s weather. So you mathematically predict a range of possible values for the outcome, and calculate an error term to qualify the certainty of your prediction.
All right, I have a little more time now. That link up above was the first thing I got when I typed “probability tutorial” into Google; I though that was amusing, so I stuck it in.
There’s a lot to say, so I’m going to concentrate on only one thing: what probability and statistics are, and how they differ.
Basically speaking, in probability, we have a theoretical model and we use that to make predictions about observations. For instance, with a six-sided die, we assume that all outcomes are equally likely, so over the long run, we should expect to see them all with roughly equal frequency.
In statistics, we work the other way around. We start with a set of observations, and we try to construct a model that fits the data. From that, we can attempt to make predictions as in probability theory, and we can gauge how good our model is by how well its predictions fit.
There are also methods whereby we can make predictions without constructing a model, but that’s more advanced.
As you might suspect, you need to know some probability before you go into statistics. Most colleges will teach them in that order, with probability being a prerequisite for statistics.