### Signal or noise?

As suggested in the post

*ET, phone home*, it is in principle possible to distinguish noise from a signal, though this may not always be practically feasible. Here we use a simple example to show how nonrandomness might be inferred (though not utterly proved).

Suppose one tosses a fair coin 8 times (or we get finicky and use a quantum measurement device as described previously). What is the probability that exactly four heads will show up?

We simply apply the binomial formula C(n,x)(p^x)(q^n-x), which in this case is set at 70/2^8, for an equivalent percentage of about 27%. The probability is not, you'll notice, 50 percent.

The probability of a specific sequence of heads and tails is 2^(-8), which is less than 1 percent. That's also the probability for 0 heads (which is the specific sequence 8 tails).

Probability for 1 head (as long as we don't care which toss it occurs in) is about 3%, for 2 heads is about 10%, and for 3 heads is about 22%.

As n increases, the probability of exactly n/2 heads decreases. The probability of getting exactly 2 heads in 4 tosses is 37.5%; the probability of exactly 3 heads in 6 tosses is 31.25%.

On the other hand, the ratio of flips to heads tends to approximate 1/2 as n increases, as is reflected by the fact that the case heads occurs n/2 times always carries the highest probability in the set of probabilities for n.

That is, if there are a number of sets of 8 trials, and we guess prior to a trial that exactly 4 heads will come up, we will tend to be right about 27% of the time. If we are right substantially less than 27% of the time, we would suspect a loaded coin.

Yet let us beware! The fact is that

*some*ratio must occur, and n/2 is still most likely. So if n/2 heads occurs over, say 100 tosses, we would not be entitled to suspect nonrandomness -- even though the likelihood of such an outcome is remote.

*Note added July 2007:*As de Moivre showed, Stirling's formula can be used to cancel out the big numbers leaving (200pi)

^{0.5}/(100pi), which yields a probability of close to 8 percent. For 1000 tosses, the chance of exactly 500 heads is about 2.5 percent; for 1 million tosses, it's about 0.08 percent.

(Suppose we don't have a normal table handy and we lack a statistics calculator. We can still easily arrive at various probabilities using a scientific calculator and Stirling's formula, which is

n! ~ (n/e)

^{n}(2n * pi)

^{0.5}

Let us calculate the probability of exactly 60 heads in 100 tosses. We have a probability of

(100/e)

^{100}(200pi)

^{0.5}/[2

^{100}(60/e)

^{60}(40/e)

^{40}2pi(2400)

^{0.5}]

which reduces to

**50**

^{100}/(40^{40}* 60^{60}) * [(200pi)^{0.5}/(2pi(2400)^{0.5}]We simply take logarithms:

x = 100ln50 - (40ln40 + 60ln60) = -2.014

We multiply e

^{-2.014}by, 0.814, which is the right-hand boldface ratio in brackets above, arriving at 0.01087, or 1.09 per cent.)

**Second thought**

On the other hand, suppose we do a chi-square test to check the goodness-of-fit of the observed distribution to the binomial distribution. Since the observed value equals the expected value, for x/500 + y/500 (that is x=0 and y=0), then chi-square equals 0.

That is, the observed distribution perfectly fits the binomial curve.

But what is the probability of obtaining a zero value for a chi-square test? (I'll think about this some more.)

*Note added July 2007:*In these circumstances, this doesn't seem to be a fair question.

**Back to the main issues**

Suppose a trial of 20 tosses was reported to have had 14 heads. The probability is less than 4% and one would suspect a deterministic force -- though the probabilities alone are insufficient for one to definitively prove nonrandomness.

Similarly, when (let's say digital) messages are transmitted, we can search for nonrandomness by the frequencies of 0 and 1.

But as we see in the

*ET*post, it may become difficult to distinguish deterministic chaos from randomness. However, chaotically deterministic sequences will almost always vary from truly random probabilities as n increases.

**Addendum**

Sometimes data sets can be discovered to be chaotic, but nonrandom, by appropriate mappings into phase space. That is, if an iterative function converges to a strange attractor -- a pattern of disconnected sets of points within a finite region -- that attractor can be replicated from the data set, even though an ordinary graph looks random.