Possibility and Probability

This weekend I finished reading the memoirs of Dr. Edward Teller
(he helped invent the atomic and hydrogen bombs). In his later years Dr. Teller
was involved with a lot of foundations to try and help get people into the
field of "Applied Science" (think hands-on practical science instead of
pure theory)

He recounted an interesting story (pg. 487) about one of the interview
questions he used to ask applicants: If you flipped a coin and each
time bet a penny on the outcome of the flip (i.e. 1 penny it would land on
heads, etc.), how much money would you have at the end of 1,000 tosses?

He said a lot of people had trouble with this one. The answer is close
to 0. Why? Each toss of the coin has a 50% chance on landing on heads
(or tails). If you toss the coin enough times, you will start to see
that you have tossed almost an equal amount of heads and tails. Betting
a penny on each toss means that you would come out almost even in the
end.

Why almost even? Each coin toss has nothing to do with any past or
future toss. It is possible to toss 100 tails in a row, but this is
unlikely. (i.e. it has a low probability) 10 in a row is possible, as
is 3, and each of those is more likely than the previous one. So when you
get to the 998th toss, if you have tossed exactly 500 heads and 498
tails, it is possible you could flip 2 more heads in a row giving you a
net profit of 2 cents (assuming you bet on heads). Neat huh?