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I am a research scientist. This is not homework.

I do not have enough points to post pictures. To see the relevant plots, see this stackoverflow question: Add 95% confidence limits to cumulative plot

On the page referenced are two plots that show coin-toss data. One shows some short runs spliced together. The other shows 100,000 tosses.

Now, I am not a mathematician (I can barely count), but it seems to me that if you link together short runs, as in the combined plot of multiple short runs, you are more likely to get crossings over the 95 percent confidence limits (the blue line), than with the 100,000 plot (where to cross the 95 percent limits is much, much harder).

Is my intuition correct?

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"I am not a mathematician (I can barely count)", I don't know many mathematicians that actually can count more than $1,2,3,\ldots,n,\ldots$ – Asaf Karagila Jun 8 '11 at 9:47
I don't know if the implication is that mathematicians can count or not. More that I am as far from a mathematician as perhaps somebody can be? – Frank_Zafka Jun 8 '11 at 10:10

At any particular point, the probability of being outside the 95% confidence interval is, by definition, 5% (though there are some minor rounding issues in the extremely short term), and this remains the same throughout the process.

So the short-term and long-term probabilities are in that sense the same. You can see this in the first chart in the accepted answer to the question you linked to, where a similar grey area lies outside the red limits.

In fact, it is almost certain that, at some stage in the extremely long term, the path of the random walk will cross the confidence limits. This is made more formal in the law of the iterated logarithm.

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Ah, well that was my alternative conclusion. But I wanted an informed opinion. Thanks for the response. – Frank_Zafka Jun 8 '11 at 10:43

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