Content deleted Content added
m Archiving 1 discussion(s) from Talk:Probability theory) (bot |
manually archive unsigned text, add unsigned notices |
||
Line 14:
Why is this page so biased towards Bayesian statistics? [[User:INic|INic]] 12:08, 19 October 2005 (UTC)
== sequences? ==
in the explanation of sample space, shouldn't the word "sequence" be "combination" as the order of the Californian voters does not matter? <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/63.237.83.52|63.237.83.52]] ([[User talk:63.237.83.52#top|talk]]) 07:59, 21 November 2005 (UTC)</small>
== 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:
[http://www.saliu.com/theory-of-probability.html]
The article is accompanied by free software pertinent to probability (combinatorics and statistics as well).
Ion Saliu,
Probably At-Large <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/66.59.117.95|66.59.117.95]] ([[User talk:66.59.117.95#top|talk]]) 15:39, 3 June 2006 (UTC)</small>
== Proposed with probability axioms ==
Line 80 ⟶ 96:
:: Good idea Hirak. I like the depth that you chose for the law of large numbers. Ideally, since the introduction also mentions the central limit theorem, that could have a similar overview-like section? But, on a completely selfish note, I don't understand how the mgf and cf and cgf relate to one other (if at all?), and I'd be thrilled to see an equally good overview here :) Maybe I should just read those articles though ;) Cheers. [[User:MisterSheik|MisterSheik]] 16:19, 29 March 2007 (UTC)
== 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 ''N''<sub>H</sub> be the number of times the coin lands heads, then we can, for any ''N'', consider the ratio <math>N_H \over N</math>.
As ''N'' gets larger and larger, we expect that in our example the ratio <math>N_H \over N</math> will get closer and closer to 1/2. This allows us to "define" the probability <math>\Pr(H)</math> of flipping heads as the [[limit (mathematics)|limit]], as ''N'' approaches infinity, of this sequence of ratios:
:<math>\Pr(H) = \lim_{N \to \infty}{N_H \over N} </math>
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'',
:<math>\left| \Pr(H) - {N_H \over N}\right| < \epsilon</math>
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. <!-- Template:Unsigned --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:MisterSheik|MisterSheik]] ([[User talk:MisterSheik#top|talk]] • [[Special:Contributions/MisterSheik|contribs]]) 16:21, 29 March 2007 (UTC)</small>
== I'm happy ==
| |||