# Checking Whether A Coin Is Fair - Estimator of True Probability

Estimator of True Probability

 The best estimator for the actual value is the estimator . This estimator has a margin of error (E) where at a particular confidence level.

Using this approach, to decide the number of times the coin should be tossed, two parameters are required:

1. The confidence level which is denoted by confidence interval (Z)
2. The maximum (acceptable) error (E)
• The confidence level is denoted by Z and is given by the Z-value of a standard normal distribution. This value can be read off a standard score statistics table for the normal distribution. Some examples are:
Z value Confidence Level Comment
0.6745 gives 50.000% level of confidence Half
1.0000 gives 68.269% level of confidence One std dev
1.6449 gives 90.000% level of confidence "One Nine"
1.9599 gives 95.000% level of confidence 95 percent
2.0000 gives 95.450% level of confidence Two std dev
2.5759 gives 99.000% level of confidence "Two Nines"
3.0000 gives 99.730% level of confidence Three std dev
3.2905 gives 99.900% level of confidence "Three Nines"
3.8906 gives 99.990% level of confidence "Four Nines"
4.0000 gives 99.993% level of confidence Four std dev
4.4172 gives 99.999% level of confidence "Five Nines"
• The maximum error (E) is defined by where is the estimated probability of obtaining heads. Note: is the same actual probability (of obtaining heads) as 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:

where n is the number of trials (which was denoted by N in the previous paragraph).

This standard error function of p has a maximum at . Further, in the case of a coin being tossed, it is likely that p will be not far from 0.5, so it is reasonable to take p=0.5 in the following:

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

Solving for the required number of coin tosses, n,