ABSTRACT
If a fair coin is tossed, the probability is 12 that it will land on its head (or tail). If the coin is tossed N times, every specific sequence of heads and tails will have probability 2−N . There are
) individual sequences
of length N that have exactly r heads and N − r tails, each one of them contributing probability 2−N . Therefore probability of obtaining exactly r heads in N tosses is
P(N, r) =
( N
r
) 2−N . (1)
While we can not predict the outcome of a single toss of a coin, if we repeat it a large number times we do expect that heads and tails will occur roughly an equal number of times. Of course we can not really expect that the number of heads and tails to be exactly equal. There will be deviations or fluctuations. If we toss the coin N times the number of heads (or tails) that we expect to have is N2 . If X is the actual number obtained X =
where Y is the deviation from the expectation. We saw that the probability that X = r is calculated easily and is given by (1)
The magnitude of the fluctuation is √
N and we can calculate asymptotically as N → ∞ the probability that Y = X − N2 is in the interval[ x1 √
N, x2 √
N ]
P(N, r) =
√ 2
π
e−2y 2 dy (2)
which is the central limit theorem. Deviations from N2 that are larger than√ N in magnitude have probabilities that go to 0 with N. For example if α > 12 and x > 0,
P
[ X ≥ N
2 + x Nα
] → 0
as N → ∞. If 12 < α < 1 we can show that for x > 0,
P
[ X ≥ N
2 + x Nα
] √
π
e−2y 2 dy = exp
[ −2x2N2α−1 + o(N2α−1)
] .
But if α = 1, the answer is different and
