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
Famous quotes containing the words true and/or probability:
“I have not yet learned to live, that I can see, and I fear that I shall not very soon. I find, however, that in the long run things correspond to my original idea,—that they correspond to nothing else so much; and thus a man may really be a true prophet without any great exertion. The day is never so dark, nor the night even, but that the laws at least of light still prevail, and so may make it light in our minds if they are open to the truth.”
—Henry David Thoreau (1817–1862)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)