Probabilty and sample spaces Definition

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In summary, the conversation discusses different ways to define probability, including using empirical probability, subjective probability, and axiomatic probability. Axiomatic probability relies on four axioms, but does not provide an explicit answer for the probability of any event. The participants also discuss the need for events to be equally likely in order to accurately determine their probabilities, and suggest using experimentation as a way to determine relative probabilities.
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ak416
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Ok, I am taking a stats course right now and I am trying to understand exactly how probability is defined. It says in the textbook that there are a few ways it can be defined. I understand the first one: Assume an experiment with n possible outcomes, each equally likely. If some event is satisfied by m of the n, then the probability of that event is m/n. However, if the events are not all equally likely, then this definition can't be used. There's also the other definitions like empirical probability and subjective probability, but these don't really give you a precise answer. Then there's the axiomatic probability with 4 axioms. But all it says is
1. P(A) >= 0,
2. P(S) = 1,
3. P(A U B) = P(A) + P(B) for mutually exclusive events A and B
4. P(the union of all mutually exclusive events) = sum from 1 to infinity (P(Ai))

this still doesn't give an explicit answer for what the probability of any event A would be! Using 3, to know P(A) i would need to know P(A U B) and P(B), and to know either of those i would need to know the other probabilities.

I think the best definition is the first definition, but then there must be a way to reduce all elements of a sample space to being equally likely.

Any insight would be greatly appreciated.
 
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Suppose you have 3 indep. outcomes A, B and C where A and B are equally likely and C is twice as likely as A or B. Then you can define events C1 and C2 as each being equally likely as A or B and run the experiment using the following routine: the first time C is observed is credited to C1. The second time C is observed is credited to C2, etc.

Prob{C} can be defined as Prob(C1} + Prob{C2}.
 
  • #3
this still doesn't give an explicit answer for what the probability of any event A would be!
And it shouldn't! There are lots of possible probability measures on any given set of events.

For example, to model an ordinary coin, you would use the uniform distribution on {heads, tails}: P(heads) = P(tails) = 1/2. To model a double-headed coin, you would use the distribution where P(heads) = 1 and P(tails) = 0.
 
  • #4
o i see, so you just have to find the probability of each event relative to the others, and using the fact that the probability of the whole sample space is 1, you would be able to find the absolute probability. It still doesn't really give an answer to how you would know which events are relatively more likely than the others (unless you can reduce the sample space to a bunch of equally likely outcomes), but i guess that must be found experimentally?
 
  • #5
ak416 said:
o i see, so you just have to find the probability of each event relative to the others... but i guess that must be found experimentally?
The definition you had posted had made an axiomatic determination of the relative probabilities ("they are all equal"); my thought experiment was an extension of that axiomatic statement.

Now you are going one step beyond the original post and asking "how did they know all outcomes were equally likely?" My guess is they could've formulated it as a hypothesis then statistically tested it; so it could've been experimental in that respect. But they did not need to. One can start with an axiomatic statement and get an aximatic definition without being experimental.
 

FAQ: Probabilty and sample spaces Definition

What is the definition of probability?

The definition of probability is the measure of the likelihood that an event will occur. It is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

How is probability calculated?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is known as the probability formula: P(A) = # of favorable outcomes / # of total outcomes.

What is a sample space?

A sample space is the set of all possible outcomes of an event. It can be represented in a list, table, or diagram to help visualize all the possible outcomes.

How does sample size affect probability?

The larger the sample size, the more accurate the probability calculation will be. This is because a larger sample size reduces the impact of random chance and outliers on the results.

How is probability used in real life?

Probability is used in many real-life situations, such as predicting the weather, analyzing stock market trends, and making decisions in fields like medicine and engineering. It is also used in gambling and games of chance to determine the likelihood of certain outcomes.

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