In a previous article, Re: Coin Probability 12/31  (posted on January 16),
it was stated that:

"You must flip the coin 100 times for it to equal out 50 heads and 50
tails."

Subsequent statements in the post led me to believe the poster meant that
literally, but that's not quite correct either -- there's nothing magical
about "100" as opposed to "10" and the coin doesn't know after 99 flips that
it must come up heads to make things even out!

If you flip a coin 100 times, you certainly expect ABOUT 50 heads but not
necessarily exactly 50 heads. In fact, anywhere from 40 to 60 heads in 100
tosses is not that surprising and quite consistent with a fair coin. A
statistician can tell you (note: some very slight algebra ahead -- and
"sqrt" means "square root"):

if you toss a fair coin N times, you will usually get between N/2 - sqrt(N)
and N/2 + sqrt(N) heads (this is "two standard deviations from the mean" for
statisticians -- it doesn't have to happen everytime, either, but it should
only fail *roughly* one time in twenty. )

Applying this formula to various values of N (for very small values of N
it's only approximate, but still reasonably close):

Toss a coin 16 times, you'd expect 8 heads, but anything between 4 and 12
isn't that surprising... note that's between 25% and 75% heads!

Toss a coin 64 times, you'd expect 32 heads, but anything between 24 and 40
isn't that surprising... note that's between 37.5% and 62.5% heads!

Toss a coin 100 times, you'd expect 50 heads, but 40 to 60 is
unsurprising... 40% to 60% heads
Toss a coin 10,000 times you'd expect 5,000 heads but 4900 to 5100 is
unsurprising 49% to 51%
etc.

There's a nice pattern here -- as N increases, the "unsurprising range" gets
bigger and bigger, but AS A PERCENTAGE OF N it gets smaller.

Ted Alper
math teacher & pokemon player.
castma@pacbell.net