What's the difference between the fields of probability and statistical inference?

A coin tossing series produces the results HHHHH. Does this provide strong evidence that the coin is not fair?

The solution is simply "Yes". I find this ridiculous, because if starting with an H, the P value of having another 4 Hs in a row is as much as 6%. The result suggests that the coin is not fair, but does not provide strong evidence that it's not. One will have to carry out a few more series in my opinion.

Do you agree with me, or am I wrong?

 

If I get delt four aces, is the deck unfair? If I win the lottery, is the draw unfair?

I don't think five data points constitutes a valid sample for this experiment.

I'm no statistician, though. I could be wrong.

 

Odd things happen rarely, but that implies they still happen by chance.

As for the strength of proof, 2 and 3 standard deviations (95% and 99%, respectively) are often considered "statistically significant" and "statistically highly significant." I tend to agree with your analysis of the coin toss. 1/2^5 is about 3%.

However, one must be careful in how the question is asked. If it is, "what is the probability of flipping a coin 5 times and having it come up the same," the answer is only 1/2^4; however, if the question is, "what is the probability it will come up heads 5 times, then the probability is 1/2^5." One must also consider how many times the trial was repeated. That is, the random probability of getting 5 heads is much higher, if you do the trial 10 times.

I think you clearly understand those principles from the way you phrased your question. So, basically, the issue is whether a 3% (97%) probability is highly significant. I don't think it is.

Finally, to put another wrinkle in the question, Richard Feynman used to demonstrate how coin flipping was biased and depended on starting position and the flipper's skill. He could reportedly get heads (or tails) as many times as he wanted.

John

 

If I get dealt four aces, is the deck unfair? If I win the lottery, is the draw unfair?

Yes! It is most definitely unfair! I should be getting the wins, not you!

--Rich

 

Do you agree with me, or am I wrong?

Personally, I agree with you, but it's a matter of judgement, unless someone specifies a probability value that defines what "strong evidence" means.

If you pick up a coin, and throw it only 5 times and get all heads, you could argue that there is strong evidence that it is a double headed coin. It would certainly warrant a quick look if you were going to gamble with it.

However, if you throw a coin many times and see one place where 5 heads showed up, but on average there is an equal number of heads and tails, then obviously there is no longer strong evidence.

This is kind of a tricky question because we intuitively want to factor in the fact that we rarely see double headed coins. However, if you start the problem with a 50 % chance that the coin is double headed, then it is much easier to say that there is strong evidence that the coin is not fair.