Understanding Probability Spaces

Question

I was looking at the following description of probability spaces, and was trying to understand this better:

In the example of flipping a coin, is the following correct?

The sample space : Heads or Tails
The event space: Some combination of outcomes you interested in observing (e.g. Heads, Heads, Tails)
The probability function: In this example, the binomial probability distribution function

Is my understanding correct?

Thanks

Reference: https://en.wikipedia.org/wiki/Probability_space

Does this answer your question? What is the formal definition of "probability distribution"? — Mittens, Oct 12 '21 at 01:57
No that’s not perfectly correct. I think you should use a more substantial experiment as your example, such as tossing a coin three times in a row. — littleO, Oct 12 '21 at 01:57
thank you everyone for your replies - I am reading the link that Oliver Diaz posted! — stats_noob, Oct 12 '21 at 01:58
Question similar to yours have been addressed before in MSE. I already sent a link to one that. You can find many more answers that may more satisfactory to you using Aproach0 engine — Mittens, Oct 12 '21 at 02:00
OP is asking a specific question about a specific example, though; we should actually answer the specific question. — littleO, Oct 12 '21 at 02:12
If you want $HHT$ to be an outcome or event, you may need to start with $\Omega={H,T}^3$ and $\mathcal F=\mathcal P(\Omega)$ — Henry, Oct 12 '21 at 22:43

score 3 · Answer 1 · answered Oct 12 '21 at 02:16

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You are correct about the sample space. We can take the sample space to be $\Omega = \{ H, T \}$.

The event space $\mathcal F$, however, should be the collection of all subsets of $\Omega$. In other words: $\mathcal F = \{ \emptyset, \{H\},\{T\},\{H, T \} \}$.

The probability function $P$ takes as input an event and returns as output the probability of that event. So, assuming the coin is fair: \begin{align} P(\emptyset) &= 0,\\ P(\{H\}) &= \frac12,\\ P(\{T\}) &= \frac12,\\ P(\{H, T \}) &= 1. \end{align}

answered Oct 12 '21 at 02:16

littleO

51,938

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However this is basically never how probability theory is actually done in practice, so working it out is really not very instructive. – Ian Oct 12 '21 at 02:27
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But if one is not able to work it out, that represents a genuine lack of understanding of the definition of a probability space. Part of learning to understand the definition of a probability space is thinking about what the probability space would be in simple examples like this. – littleO Oct 12 '21 at 02:39
In a way I disagree, I find these "build a suitable sample space" exercises are misleading to how people actually do things. It is more useful to have a strong sense of the things that can be done within the framework (e.g. iid sequences are possible, continuous time white noise is not) and treat how you get that back down to foundations as an enrichment exercise to me. – Ian Oct 12 '21 at 13:33
This is not a criticism of the answer (it answers the question that was asked). It is more a warning to the OP that this question might be less useful than they think it is. – Ian Oct 12 '21 at 13:56
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I liked your answer. And I think it segues well into this more extensive post in Cut The Knot by the late Alexander Bogomolny. – Antoni Parellada Oct 12 '21 at 23:22

Understanding Probability Spaces

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