# Talk:Probability theory

## sequences?

in the explanation of sample space, shouldn't the word "sequence" be "combination" as the order of the Californian voters does not matter?

## Don't use F to mean two completely different things

I'm going to remove all references in this page to F as an event. It will only confuse people later when they see that F is a sigma-algebra. E=>A and F=>B is my suggested universal fix.

I concur. Additionally, it would be nice for Wikipedia articles to have consistency. Hence, I would also make the suggestion to use the script F that Probability space uses: "A probability space is a mathematical triplet ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ or to use standard sigma algebra notation ${\displaystyle (\Omega ,\Sigma ,P)}$. Kdmckale (talk) 00:00, 29 September 2015 (UTC)

## Probability is related to life

The article on probability theory is superficial. It uses jargon, while being disconnected from real life. I believe that the best foundation to theory of probability is laid out here:

The article is accompanied by free software pertinent to probability (combinatorics and statistics as well).

Ion Saliu, Probably At-Large

## Now almost totally redundant, unless someone wants to merge something back in

To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor. Certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible, for example, to flip 10 heads in a row. What then does the number "50%" mean in this context?

One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent—that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let NH be the number of times the coin lands heads, then we can, for any N, consider the ratio ${\displaystyle N_{H} \over N}$.

As N gets larger and larger, we expect that in our example the ratio ${\displaystyle N_{H} \over N}$ will get closer and closer to 1/2. This allows us to "define" the probability ${\displaystyle \Pr(H)}$ of flipping heads as the limit, as N approaches infinity, of this sequence of ratios:

${\displaystyle \Pr(H)=\lim _{N\to \infty }{N_{H} \over N}}$

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,

${\displaystyle \left|\Pr(H)-{N_{H} \over N}\right|<\epsilon }$

In other words, by saying that "the probability of heads is 1/2", we mean that if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitrarily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.

Note that a proper definition requires measure theory, which provides means to cancel out those cases where the above limit does not provide the "right" result (or is even undefined) by showing that those cases have a measure of zero.

The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz & Guildenstern Are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event—after all, it is possible (although unlikely) that a fair coin would give this result—or whether his assumption that the coin is fair is at fault.

## Statistics or Mathamtics

The article claims that Probability Theory is a branch of mathematics. However, probability is fundamental to understanding data (statistics), just as arithmetic is fundamental to understanding mathematics, so it should be more appropriate to say that probability is a branch of statistics. Moreover, several US colleges consider Probability to be a subject within statistics, and are taught through the statistics department.--104.38.180.88 (talk) 18:28, 29 March 2018 (UTC)