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:
- The confidence level which is denoted by confidence interval (Z)
- 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,
Read more about this topic: Checking Whether A Coin Is Fair
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