The Effect of Perspective on Probability My friend and I are tearing each other to bits over this, hope someone can help.
Coin flip experiment:
Define a single trial as 10 coin flips of a fair coin. Perform an arbitrarily large number of trials. At some number of trials n, you notice that your distribution is extremely skewed in one direction (i.e., the "average" of your 10-flip sets is far away from 5 heads and 5 tails).
My reaction: Because you are guaranteed to hit a 5H/5T mean as n approaches infinity, the probability that the next n trials contains an equal skew in the opposite direction increases. In other words, given 2*n* trials, if the first n are skewed in one direction, than the remaining n are probably skewed in the other direction such that the overall distribution of your 2*n* trials is normal and centered around 5H/5T.
My friend's reaction: It doesn't matter if your first n trials is skewed, the next n trials should still represent an unmodified 5H/5T distribution regardless. The probability of the next n trials being skewed in the opposite direction is unchanged and low.
Who's right, and why?

If you want to consider something really interesting, consider that as you add more flips, the probability of getting exactly half heads and tails goes down and this isn't that hard to compute mathematically as the sequence is the fraction of $2n \choose n$ divided by $(2n)^2$ for 2n flips.  The denominator will grow as you increase the number of flips.
For 2 flips, this is 2/4 = 1/2.
For 4 flips, this is 3/8.
For 6 flips, this is 20/64 = 5/16.
Now, as this continues, it will keep diminishing as it is a specific outcome compared to be "near" the middle.  Just something to consider as while you may get close to the middle, the exact middle keeps having lower and lower probability.
