Talk:Checking whether a coin is fair

message: "The graph on the right shows the probability density function of r" - Is this really the case? Isn't the area under the curve in a PDF always 1? This doesn't seem to be the case in the graph.  — Preceding unsigned comment added by 213.142.96.115 (talk) 11:59, 22 July 2020 (UTC)
replies: text: That indeed is a PDF. The integral from 0 to 1 (the entire probability space) of that function is 1 as per WolframAlpha. 78.1.182.236 (talk) 16:48, 18 April 2022 (UTC)
thread_title: PDF

message: To check the bias of the coin, given the event of h heads out of n tosses, one has to calculate the probability of such an event to occur given that the coin is fair. Then one has to decide whether the coin is fair based on whether such a probability is acceptable or not. The probability of the event, given the fair coin, is equal to
: Pr\{|H-N/2|\geq\epsilon\}=2\sum_{i=n/2+\epsilon}^n {n \choose i}0.5^i\cdot0.5^{n-i}
where \epsilon is the observed deviation. The formula accounts for possible deviation either way as well as deviations larger than observed. For example, if one observes 7 heads out of 10 tosses, a probability of such a deviation from expected value of .5 is equal to
:2\sum_{i=7}^{10} {10 \choose i}.5^{10}\sim 34.38%
Then one has to decide whether such probability is acceptable. In the example above, the event is not uncommon, so the coin can be fair. The fact that the observation may be more probable given a somewhat unfair coin is irrelevant. 
(Igny 18:12, 19 September 2005 (UTC)
thread_title:  Correct answer 

message: I've gone through and changed most of the language that made this seem like a how-to guide, and removed the "candidate to be moved to Wikibooks" notice accordingly. I got rid of all the necessary "we"'s, "you"'s, "one"'s, et cetera in favor of more straightforward wording.
However, I can't say I've improved the factual accuracy of the article significantly (and I don't have time to research the material in question at the moment), so I've left the corresponding notice up for now. - 68.20.21.191 04:53, 15 Apr 2005 (UTC)
thread_title:  "How-To" Format Changes/Candidacy for Movement 

message: Do we need both this article and Checking if a coin is biased
Blaise 09:10, 28 Apr 2005 (UTC)
replies: text: What a mess. Those are two incompatible forks of coin flipping. Merging in the changes from ...biased will be "fun". I added a notice to that article, because the present article contains more material and was edited more recently. --MarkSweep 16:35, 28 Apr 2005 (UTC)
replies: text: Sure. So does mean we go into "merge into" and "merged with" tags? I think this page should be the final page. But they are more or less the same, I say merge. HereToHelp 12:37, 13 October 2005 (UTC)
thread_title: Mergers and acquisitions

message: I've moved this cleanup tag from the article:
This article needs to be cleaned up to conform to a higher standard of quality.  See Wikipedia:How to edit a page and :Category:Wikipedia help for help, or this article's talk page.
While the article could certainly be improved, it's not obvious to me why this tag is warranted. Enchanter 21:28, 30 October 2005 (UTC)
thread_title:  Cleanup tag 

message: Jaw-droppingly unencyclopedic. This thing needs deletion. --⁪frothT C 07:45, 29 December 2006 (UTC)
replies: text: I thought it was a pretty good example of how significance testing can be applied to real life. --M1ss1ontomars2k4 (T | C | @) 01:26, 7 February 2007 (UTC)
thread_title:  Wow. 

message: :Moved here from User talk:Trialsanderrors
I've moved checking if a coin is fair back to that title (and the title perhaps could be improved) and created a new article titled fair coin.  The latter topic is much broader than the narrower topic of checking if a coin is fair.  Possibly fair coin should redirect to Bernoulli trial; I will think about that.  One goes through graduate school reading incessantly about (metaophorically named) "fair coins" and one reads innumerable scholarly papers relying on the concept of "fair coins"; and the topic is not ONLY about statistical hypothesis testing treating only that one hypothesis. Michael Hardy 20:35, 19 February 2007 (UTC)
replies: text: Well "Checking if a coin is fair" has no encyclopedic standing, since the title itself implies a how-to. The article needs to expand on the statistical concenpt of a fair coin and the discrepancy between theoretical construct and real world application, but it should be done in conjunction. ~ trialsanderrors 21:03, 19 February 2007 (UTC)

text: Your definition at fair coin is incorrect too. A fair coin is a Bernoulli random variable with p = 0.5, a Bernoulli trial is the coin toss, and a Bernoulli process is the series of coin tosses. ~ trialsanderrors 06:15, 20 February 2007 (UTC)
thread_title: Fair coin

message: At the top of the "Estimator of True Probability" section, there's a little box with an equation in it.  Not being a statistician, I have no idea what the "t" in it is supposed to be.  There is seemingly no reference to it in the article... 65.183.135.40 (talk) 03:40, 18 February 2008 (UTC)
replies: text: See first paragraph of the previous section. h and t are the numbers of heads and tails respectively.77.73.111.214 (talk) 14:48, 19 February 2008 (UTC)
thread_title:  What the heck is "t"? 

message: Yet another problem here is that the usual standard error formula is not a very good one for estimating binomial proportions.  See e.g. 
Plf515 (talk) 15:56, 13 March 2008 (UTC)
thread_title:  Problem with standard error formula 

message: Breaks down when the article starts speaking about priors without saying what they are or linking to their definition. Furthermore, saying that $r$ is a probability, and then using the notation $\mathrm{Pr}(event)$ doesn't help. I just stopped reading, what are events and what are probabilities should be absolutely clear from the notation.  —Preceding unsigned comment added by 138.246.7.147 (talk) 09:55, 14 March 2008 (UTC)
thread_title:  Understandability 

message: In the article, the example is used: "For example, let n=10, h=7, i.e. the coin is tossed 10 times and 7 heads are obtained:"
So why does the n become h+t+1? Isn't that like saying the coin is tossed 11 times? Shouldn't it be h+t? i.e. 10?
Aatombomb (talk) 22:31, 26 March 2008 (UTC)
thread_title:  Math error? 

message: Yes... I've seen some older discussions about it above. Still this article goes into a bit too extensive detail (it gets too close to school manual style, I think), while the other is a bit poor. So I think merging would help. - Nabla (talk) 23:34, 7 May 2008 (UTC)
replies: text: This last point (regarding "Checking_if_a_coin_is_biased") has evidently been dealt with, as that name now redirects to this article. Melcombe (talk) 12:35, 11 January 2010 (UTC)
replies: text: I have removed the merge template, as no action taken and earlier discussion of difference in context for fair coin seems decisive. Melcombe (talk) 13:20, 11 January 2010 (UTC)
thread_title:  Merge to Fair coin 

message: I have added this tag because:
* as noted above, the Bayesian section does not well explain is arguments at all well.
* the topic of estimating the probability applies equally to both Bayesian and frequentist approaches, but is apparently only in the frequentist bit
* there was an attempt to retitle the main subsections as Bayesian and frequentist, but this requires some rewriting of the starting portions so that they make sense having done this
*much of the "frequentist" bit seems taken up with the question of finding a sensible sample size, but this should be achievable in a Bayesian context as well
*the Bayesian bit should at least point to the conjugate distributions and related results.
Melcombe (talk) 12:15, 6 July 2009 (UTC)
thread_title:  Expert tag July 2009 

message: The text says "The true probability of obtaining a particular side when a fair coin is tossed is unknown, but the uncertainty is initially represented by the "prior distribution"." in the preamble. I do not understand this. Isn't the probability to get a particular side with *fair* coin 50% by definition? Given my (limited) understanding of Bayesian probability, I propose to use the following text instead:
"Initially, the true probability of obtaining a particular side when a coin is tossed is unknown, but the uncertainty is represented by the "prior distribution"." --Snipergang (talk) 09:47, 22 December 2015 (UTC)
thread_title:  Unclear preamble 

message: I was working through the examples and noticed that getting the supplied answers requires low statistical power, (1-β)=0.5, where β = p(false negative error). This doesn't make much sense; in real-world applications, the false negative rate is at least as or perhaps more important than the false positive rate α. The current example in the article is heavily weighted toward confidently detecting fair coins at the expense of missing a lot of unfair coins.
Confidence should include successfully classifying both fair and unfair coins, not just fair coins.
If you want to be totally unbiased in your confidence, you should consider α and β together with the ratio α/β=1. Confidence is how often you're correct, or C= 1-(α+β); false positive and false negative results are mutually exclusive events and thus their probabilities can be added.
These calculations can be done with G*Powerhttp://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower.html, a free software package for working with error, sample sizes, and effect sizes. 
Example 1 currently reads:
1. If a maximum error of 0.01 is desired, how many times should the coin be tossed?
:n = 2500\,  at 68.27% level of confidence (Z=1)
:n = 10000\,  at 95.45% level of confidence (Z=2)
:n = 27225\,  at 99.90% level of confidence (Z=3.3)
But really, taking α/β=1 and considering confidence = 1-(α+β), I get:
:n = 14512\,  at 68.27% level of confidence (Z=1)
:n = 45738\,  at 95.45% level of confidence (Z=2)
:n = 114609\,  at 99.90% level of confidence (Z=3.3)
--Dubium et Libertas (talk) 14:42, 15 December 2019 (UTC)
thread_title:  False negative rate is 0.5 in the examples (low power)