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?