Sometimes when choosing a coin (particularly for a coin flip), it may be desirable to determine if the coin is fair – that is, if the probability of obtaining a given side (commonly heads or tails) in the toss is 50%.

Preamble

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This article is about how to determine if a coin is fair. There are many methods for doing this and this article describes two methods for doing so.

Both methods described in this article involve performing an experiment (or trial) which involves tossing the coin many times and recording down the result of each and every toss. Based on the mathematical analysis of the result of all the tosses, a decision can then be made on whether the coin could be declared "fair".

• Posterior probability density function. This method assumes that the number of tosses is fixed and is not under the direct control of the experimenter. The method requires the probability density function (PDF) of the true value of obtaining a particular side be derived and the probability of a "fair coin" be obtained by integrating the PDF by the relevant interval. Based on result, a decision can then be made whether to proclaim the coin as a fair coin.

• Estimator of true probability. This method assumes that the experimenter can decide and implement any number of coin tosses for the particular coin. The experimenter decides on what level of confidence they require and how large the margin of error they are willing to tolerate. This then determines the minimum number of tosses that needs to be perform in the experiment.

Posterior probability density function

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One way of verifying this is to calculate the posterior probability density function of Bayesian probability theory.

A test is performed by tossing the coin n times and noting the number of heads h and tails t:

H = h (Total number of heads is h)

T = t (Total number of tails is t)

N = n = h + t (Total number of tosses is n)

Next, let r be the actual probability of obtaining heads in a single toss of the coin. This is the value desired. Using Bayes' theorem, posterior probability of r conditional on H and T is expressed as follows:

$f(r\; |\; H=h,\; T=t)\; =$

\frac {\Pr(H=h | r, N=h+t) \, f(r)} {\int_0^1 \Pr(H=h |r, N=h+t) \, f(r) \, dr}. \!

The prior summarizes what is known about the distribution of r in the absence of any observation. We could assume (but it would be rather ridiculous) that the prior distribution of r is uniform over the interval 1. That is, f(r) = 1. In fact, we ought to use a prior distribution that reflects our experience with real coins.

The probability of obtaining h heads in n tosses of a coin with a probability of heads equal to r is given by a binomial distribution:

$\backslash Pr(H=h\; |\; r,\; N=h+t)\; =\; \{h+t\; \backslash choose\; h\}\; \backslash ,\; r^h\; \backslash ,\; (1-r)^t.\; \backslash !$

Putting it all together:

$$

f(r | H=h, T=t) = \frac| < E where $p\backslash ,\backslash !$ is the estimated probability of obtaining head. Note: $p\_\{\backslash mathrm\{actual\}\}\backslash ,\backslash !$ is the same actual probability (for obtaining head) as the term $r\backslash ,\backslash !$ of the previous section in this article.

• In statistics, the estimate of a proportion of a sample (denoted by p) has a standard error (standard deviation of error) given by:

$s\_p\; =\; \backslash sqrt\{\; \backslash frac\; \{p\; \backslash ,\; (1-p)\; \}\; \{n\}\; \}$

This standard error $s\_p$ will have a maximum theoretical value if $p\; =\; (1-p)\; =\; 0.5$.

Hence , assuming the worse case , $p$ is set to 0.5 to get the maximum possible value of $s\_p$.

$s\_p\backslash ,\backslash !$	$=\; \backslash sqrt\{\; \backslash frac\; \{p\; \backslash ,\; (1-p)\; \}\; \{n\}\; \}\; =\; \backslash sqrt\{\; \backslash frac\; \{0.5\; \backslash times\; 0.5\; \}\; \{n\}\; \}$
	$=\; \backslash sqrt\{\; \backslash frac\; \{\; 1\; \}\; \{4\; \backslash ,\; n\}\; \}\; =\; \backslash frac\; \{1\}\{2\; \backslash ,\; \backslash sqrt\{n\}\}$

And hence the value of maximum error (E) is given by

$E\backslash ,\backslash !$

$=\; Z\; \backslash ,\; s\_p\; =\; \backslash frac\; \{Z\}\{2\; \backslash ,\; \backslash sqrt\{n\}\}$

Therefore, the final formula for the number of coin tosses for the estimator $p\backslash ,\backslash !$ is

$E\; =\; \backslash frac\; \{Z\}\{2\; \backslash ,\; \backslash sqrt\{n\}\}\; \backslash quad\; \backslash quad\; \backslash mbox\{or\}\; \backslash quad\; \backslash quad\; n\; =\; \backslash frac\; \{Z^2\}\; \{4\; \backslash ,\; E^2\}\; \backslash !$

provided that $n\; \backslash cdot\; p\; \backslash ge\; 5$ and $n\; \backslash cdot\; q\; \backslash ge\; 5$ where $q\; =\; (1-p)\backslash ,$ to satisfy the Central Limit Theorem.

Example

1. If a maximum error of 0.01 is desired, how many times should the coin be tossed?

$n\; =\; \backslash frac\; \{Z^2\}\; \{4\; \backslash ,\; E^2\}\; =\; \backslash frac\; \{Z^2\}\; \{4\; \backslash times\; 0.01^2\}\; =\; 2500\; \backslash \; Z^2$

$n\; =\; 2500\backslash ,$ at 68.27% level of confidence (Z=1)

$n\; =\; 10000\backslash ,$ at 95.45% level of confidence (Z=2)

$n\; =\; 27225\backslash ,$ at 99.90% level of confidence (Z=3.3)

2. If the coin is tossed 10000 times, what is the maximum error of the estimator $p\backslash ,\backslash !$ on the value of $r\backslash ,\backslash !$ (the actual probability of obtaining head in a coin toss)?

$E\; =\; \backslash frac\; \{Z\}\{\; 2\; \backslash ,\; \backslash sqrt\{n\}\; \}$

$E\; =\; \backslash frac\; \{Z\}\{\; 2\; \backslash ,\; \backslash sqrt\{\; 10000\; \}\; \}\; =\; \backslash frac\; \{Z\}\{\; 200\; \}$

$E\; =\; 0.0050\backslash ,$ at 68.27% level of confidence (Z=1)

$E\; =\; 0.0100\backslash ,$ at 95.45% level of confidence (Z=2)

$E\; =\; 0.0165\backslash ,$ at 99.90% level of confidence (Z=3.3)

3. The coin is tossed 12000 times with a result of 5961 heads (and 6039 tails). What interval does the value of $r\backslash ,\backslash !$ (the true probability of obtaining head) lie within if a confidence level of 99.999% is desired?

$p\; =\; \backslash frac\{h\}\{h+t\}\; \backslash ,\; =\; \backslash frac\{5961\}\{12000\}\; \backslash ,\; =\; 0.4968$

{| border="0" cellpadding="5" cellspacing="0" align="none" Now find the value of Z corresponding to 99.999% level of confidence.

$Z\; =\; 4.4172\; \backslash ,\backslash !$

{| border="0" cellpadding="5" cellspacing="0" align="none" Now calculate E

$E\; =\; \backslash frac\{Z\}\{2\; \backslash ,\; \backslash sqrt\{n\}\}\; \backslash ,\; =\; \backslash frac\{4.4172\}\{2\; \backslash ,\; \backslash sqrt\{12000\}\}\; \backslash ,\; =\; 0.0202$

{| border="0" cellpadding="5" cellspacing="0" align="none" The interval which contains r is thus:

Hence, 99.999% of the time, the interval above would contain $r\backslash ,\backslash !$ which is the true value of obtaining head in a single toss.

Other applications

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The above mathematical analysis for determining if a coin is fair can also be applied to other uses. For example:

• Determining the product defective rates of a product when subjected to a particular (but well defined) condition. Sometimes a product can be very difficult or expensive to produce. Furthermore if testing such products will result in their destruction, a minimum amount of products should be tested. Using the same analysis the probability density function of the product defect rate can be found.

• Two party polling. If a small random sample poll is taken where the there are only two mutually exclusive choices, then this is equivalent to tossing a single coin multiple times using a bias coin. The same analysis can therefore be applied to determine actual voting ratio.

• Finding the proportion of females in an animal group. Determining the gender ratio in a large group of an animal species. Provided that a very small random sample is taken when performing the random sampling of the population, the analysis is similar to determining the probability of obtaining heads in a coin toss.

See also

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• binomial test
• coin flipping
• margin of error
• statistical randomness
• Estimation theory
• Inferential statistics
• Point estimation

External links

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• frequentists and bayesians

Probability theoryStatistics