# Can a naive hypothesis test determine that a coin is approximately fair?

#### spectrum

##### New Member
It's a very elementary example to determine fairness of a coin with repeated tosses and a hypothesis test, at least approximately. If the result of repeated tosses is fair, the coin is classified as fair.

However, a unfair coin, C, that can equally probably switch between two unfair states, p and (1-p) for heads respectively, may pass repeated tosses and be classified as fair.

Of course C may be examined for unfair switches, but there is no amount of tosses that may separate a fair coin and C.

Does this mean that naive hypothesis tests are dubious? On what condition may hypothesis tests be credibly employed?

#### hlsmith

##### Not a robit
Well the law of large numbers states with infinite tosses the long run frequency should be 0.50 if its a fair coin. Given this, a coin in a not long run can by chance have 50 heads in a row. Though the probability is low. You can run a binomial test with a given probability for fairness and calculate the probability of getting that many heads or tails given the value you are examining.

#### spectrum

##### New Member
Well the law of large numbers states with infinite tosses the long run frequency should be 0.50 if its a fair coin. Given this, a coin in a not long run can by chance have 50 heads in a row. Though the probability is low. You can run a binomial test with a given probability for fairness and calculate the probability of getting that many heads or tails given the value you are examining.
Thanks. However, a binomial test would classify both a fair coin and the unfair coin C as fair due to the existence of switch.

#### Dason

If you use a hypothesis that assumes a constant probability of success then sure if you have a magical coin that can switch probabilities then maybe it won't detect that. We can certainly devise a test that would be able to detect that but hypothesis tests can't assume things about the random variables that you don't tell it to assume.

#### spectrum

##### New Member
If you use a hypothesis that assumes a constant probability of success then sure if you have a magical coin that can switch probabilities then maybe it won't detect that. We can certainly devise a test that would be able to detect that but hypothesis tests can't assume things about the random variables that you don't tell it to assume.
Is there a rigorous qualification for sound uses of hypothesis tests?