Counting Events in a Finite Coin-Toss Probability Space

Summary

The document asks how to interpret an expression for the number of sets in a sigma-algebra in a probability-space example. The answer explains the counting in two stages: an experiment with a finite number of coin tosses has a set of possible outcome sequences, and the sigma-algebra of all subsets of that outcome space has a cardinality that is a power of two again. This distinguishes the number of outcomes from the number of events.

The discussion is elementary and does not address trading or stochastic-calculus applications beyond the textbook context that motivated the question. One answer gives the general counting relationship, while another response speculates about the notation and expresses uncertainty about the nested exponent. Thus, the main useful point is the power-set counting principle; the alternative explanation is not a reliable extension and should not be treated as a worked derivation for every possible sigma-algebra.

Key ideas

• A finite coin-toss experiment has a finite set of possible outcome sequences.
• A sigma-algebra containing every subset of a finite outcome space has size equal to the number of its subsets.
• The number of outcomes and the number of events are different quantities.
• The discussion's speculative interpretation of the nested exponent is less dependable than the power-set explanation.

Tags

# How to count number of sets in a sigma-algebra?


# How to count number of sets in a sigma-algebra?












Here is an example for the construction of a probability space in Shreve's stochastic calculus, page 4, what is the meaning for $2^{(2^0)}$? it seems like a method to calculate number of subsets, but I still cannot fully understand.Thanks for help!

## Answer by Wei (score 2, accepted)

https://quant.stackexchange.com/a/80325

The probability space corresponding to the experiment of flipping $x$ coins has $2^x$ possible outcomes. So these outcomes generate a sigma-algebra of size $2^{2^x}$.

## Answer by KaiSqDist (score 1)

https://quant.stackexchange.com/a/80322

I am not exactly sure, but because $\Omega_\infty$ is a set of possible outcomes, the 2 sets must refer to $\emptyset$ and $\Omega$. Then, based on this, I would guess that $\textbf{2}^{2^0}$ refers to the number of sets, $2^{\textbf{2}^0}$ (the exponential two) refers to the outcomes, which are heads or tails.

Not too sure about the zero that is the exponential's exponential though.

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