The error would be in determining p, not the subsequent calculation.
You don't seem to put very much faith in math.
If the probability of flipping heads is 1/2, then the probability of flipping heads 8 times in a row is (1/2)^8 = 1/256. If you throw 8 coins on the floor, you have a 1/256 chance of getting 8 heads. In other words, If you throw them on the floor, pick them up, throw them again, pick them up, throw them again, etc. and repeat this for a few years, you will be astonished to find that you got 8 heads almost exactly once every 256 tries. The accuracy of the prediction has nothing to do with the fact that we're using 8 coins. The prediction will always be accurate. That's why we do math.
"You know that's a large underestimation"? Ha! No, I actually know it to be pretty spot on. What we've been talking about is a version that's a little more complicated since the probability isn't 1/2, but it's not really any different.