It has been suggested that this article or section be merged with Checking if a coin is fair. (Discuss)

When given a coin, we may wish to find out if the coin is fair (i.e. the probability of obtaining head (or tail) in a toss is 50%. One way of verifying this is to calculate the probability density function using Bayesian probability theory. Wikipedia does not have an article with this exact name. ... 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%. // Posterior probability density function One way of verifying this is... 1¢ euro coin A coin is usually a piece of hard material, generally metal and usually in the shape of a disc, which is used as a form of money. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... Bayes theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. ...

First perform a test by tossing the coin N times and carefully note down the number of heads H. So we have :

n = N = total number of tosses

h = H = total number of heads

T = N − H = total number of tails

Next, let r be the actual probability of obtaining head in a single toss of the coin. This is the value which we wishes to find. Using Bayesian probability theory, we have:

The prior summarizes what we know about the distribution of r in the absence of any observation. We will assume in this case that the prior distribution of r is uniform over the interval [0, 1]. That is, f(r) = 1. That assumption should be considered provisional -- if some additional background information is found, we should modify the prior accordingly.

The probability of obtaining H heads in N toss of a coin with any value of r is given by

Putting it together we have :

Now using the identity

We obtained have the final formula for the probability density function: In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

Example

For example: N = 10, H = 7 ie. We toss the coin 10 times and get 7 heads

Graph of example

Graph of y = 1320x⁷ (1 − x)³ with x ranging from 0 to 1

This graph is the graph of the probability density function of r given that we had obtained 7 heads in 10 tosses (Note: r is the actual probability of obtaining head when tossing that coin). Image File history File links graph of y=1320 * x^7 * (1-x)^3 Plotted using gnuplot. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

So is the coin biased? One can be fairly confident that the coin is indeed biased because the probability Pr(0.45 < r < 0.55) of an unbiased coin is quite small when compared with the alternative hypothesis (a biased coin).

Shape of curve

The astute person would notice that the shape of the plotted curve is solely determined by the numerator

while the denominator

determines only the scaling of the plotted curve.

This means that you can plot the shape of the curve using just the equation and by observing the plotted curve, you can ascertain whether the coin is bias and roughly how much bias it is.

The value of r where f(r) have the maximum value is r_max = H/N as you would have expected.

How many times should the coin be tossed

To determine the number of times, a coin should be tossed, you need two vital criteria.

1. The radius Z of the confidence interval
2. The maximum acceptable error (E)

• The radius of the confidence interval is denoted by Z and is the Z-value of a normal/gaussian curve.

Z = 1.0 gives 68.27% confidence

Z = 2.0 gives 95.45% confidence

Z = 3.0 gives 99.73% confidence

Z = 3.3 gives 99.90% confidence

• The maximum acceptable error is defined by where p is estimated probability of obtaining heads.

The formula for the number of coin tosses is In statistics, confidence intervals are the most prevalent form of interval estimation. ...

provided that and where to satisfy the central limit theorem.

Central limit theorem - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

Examples

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

at 68.27% confidence (Z = 1)

at 95.45% confidence (Z = 2)

at 99.90% confidence (Z = 3.3)

2. If the coin is tossed 40000 times, what is the maximum error of the estimated value of p (obtaining head)?

at 68.27% confidence (Z = 1)

at 95.45% confidence (Z = 2)

at 99.90% confidence (Z = 3.3)