Coins

Background

Because we are all familiar with them, and because they are so simple, problems about coins are very common in introductory texts about probability.

The difficulty with this is that people are so familiar with everyday coins that they invariably overlook the need to define their mathematical terms: they don't actually say what a coin is so far as their analysis is concerned. As a result, they struggle with problems which -if they were to think about definitions- would be obvious or, at any rate, obviously badly-stated.

Is "A coin" meant to be a real coin: one of the physical discs we use to buy things?: If so, then where are the statistics about the distributions of Pr(H) and Pr(T) which would normally be expected to accompany such questions?
Is "A coin" meant to be a fair coin?: If so, then what are we to make of questions such as "A coin is tossed 10 times and comes down H each time. What is the probability it will come down H on the next toss?" The answer, of course, is ½ : after all, the coin is fair. If "coin" means "fair coin" then answer will always be ½ regardless of the number of tosses; yet the anwer is rarely ½, but is usually an enjoinder to use the Law of Succession which, incidentally, does not give the probability.
Is "A coin" some form of mathematical construct?: If so, where is the definition?

Just as importantly, as the whole raison d'être of mathematical coins is that they be tossed, what do we mean by "tossing" a coin?

This page introduces my own view of sensible definitions and then investigates them further.

What is a coin?

For our purposes, a coin is any element of S(2).That is, it is a pair of strictly positive numbers summing to 1, eg. (0.6, 0.4), (0.01, 0.99) etc. [ S(2) is the open-ended straight line segment joining (1,0) and (0,1) ] By convention, we use the labels ("H","T") rather than ("1","2").

There is no such thing as a double-headed, or double-tailed, coin; for if there were then they would be (1,0) and (0,1) -which are not elements of S(2).
The coin (½,½) is called the fair coin.
A physical disc used for the purposes of trade will be called a minted coin. From the mathematical point of view, the set of minted coins might be approximated by a small-diameter set of coins. The meaning of small is a matter for experimental investigation. Likewise, the centre of the set would also be a matter for experimentation but, in the absence of such experiments, could be taken as the fair coin.

What is a toss of a coin?

A 'toss' of the coin f is defined algorithmically.

We use the computer's RANDOM function to select a number in ]0,1].

If that number is in ]0,f(1)] we say that we have 'tossed "H" ', or that the coin has 'come down "H" ';
if it is in ]f(1),f(1)+f(2)], =]f(1),1], that we have 'tossed "T" ', or that the coin has 'come down "T" '.

definition of a coin

The probability of "H", given f, is the likeliness over the singleton set {f} of "H". This is f(1), so the probability of "H" is f(1), and the probability of "T" is f(2).

Bias

The coin f is biased towards H if f(1)>½, ie. if f(1)>f(2). It is biased towards T if f(2)>½, ie. if f(2)>f(1). A coin is biased if it is biased towards H or biased towards T.

No coin is both biased towards H and biased towards T.: For if f were such a coin then the bias towards H would mean that f(1)>½, and the bias towards T would mean that f(2)>½ so that f(1)+f(2)>½+½, ie f(1)+f(2)>1, contradicting f(1)+f(2)=1.
The fair coin is the only coin which is not biased.: We firstly note that the fair coin is neither biased towards H nor biased towards T, and so is not biased. Now let f be any coin which is not biased. Then f is neither biased towards H nor biased towards T. Since f is not biased towards H it must be that f(1)≤½ and since f is not biased towards T it must be that f(2)≤½. Since both f(1)≤½ and f(2)≤½ then f(1)+f(2)≤1, with equality iff they are both ½, that is iff f =(½,½) and so is the fair coin.

S(2)

Since the fair coin is the only coin which is not biased

Saying "Assuming the coin is not biased" is equivalent to saying "Assuming the coin is the fair coin"
The set of unbiased coins is {(½,½)}, which is singleton and so has measure zero. So the answer to the question "Is the coin biased?" will almost always(ie, except on a set of measure zero) be "Yes", regardless of any data.

Probabilities and likelinesses

Reminder: a probability is a likeliness over a singleton set

Multinomial Theorem
Underlying set: {(½,½)}
Number of tosses	Number of H
Number of tosses	0	1	2	3
0	1
1	1	1
2	1	2	1
3	1	3	3	1

If the underlying set is the singleton set {f} and g is some integram then the likeliness of g is a probability and is given by using the Multinomial Theorem as M(g)f^g. If f is the fair coin, we use Pascal's triangle to display the relative proportions of the likelinesses (probabilities) of the various integrams (see left).

Combination Theorem
Underlying set: S(2)
Number of tosses	Number of H
Number of tosses	0	1	2	3
0	1
1	1	1
2	1	1	1
3	1	1	1	1

If the underlying set is S(2) then the likeliness of g given 0 is obtained from the Combination Theorem. We use another triangle to display the relative proportions of the likelinesses of the various integrams (see right).

If we are dealing with a problem which says that "a fair coin is tossed" then we use the underlying set {(½,½)}. If the problem states merely that "a coin is tossed" then the appropriate underlying set would be S(2): except that we need to make sure that whoever is posing the problem understands what s/he is asking for, and is not -for example- thinking of a minted coin but expressing themselves badly.

Transition between Multinomial and Combination Theorems

If we are modelling a minted coin then we need to use an underlying set (centred on the fair coin) with a small diameter. To do this, we use a contraction of S(2) centred on the fair coin. Since we do not know which diameter (magnitude) to use, we shall consider a range of values.

If we use a magnitude of 0 then the underlying set becomes {(½, ½)} and the Multinomial Theorem applies. If we use a magnitude of 1 then the underlying set remains as S(2) and the Combination Theorem applies. By varying the magnitude of the contraction from 0 to 1, we can observe the transition from the Multinomial Theorem to the Combination Theorem.

In the diagram below, we are modelling two tosses of a coin, so there are three possible results: (2,0), (0,2) and (1,1). Notice how the curves remain almost horizontal for magnitudes less than about 0.3. This means that in practice it would be very difficult to detect bias within the limits (0.35,0.65) by using two tosses: the limits, here, are 0.5±0.3*0.5.

If we are not trying to model a minted coin then we do not need to centre on the fair coin. In the next diagram, the centre is (0.95,0.05). Notice how the likeliness of (1,1) reaches a maximum at a contraction of magnitude about 0.7, ie not at either end.

In both diagrams, the three curves meet at the value ⅓. Calculations were carried out using my program Great Likelinesses.

Boys and girls

What do children have to do with coins? They ARE coins. To be precise, they are tosses of the fair coin.

We represent the outputs by the symbols ("boy","girl") rather than ("H","T") or ("1","2").

The gender of a child is determined by tossing the fair coin once. Two or more tosses give the genders of the appropriate number of children: the first toss is the eldest child, the last toss is the youngest, and so on.

Since the set of fair coins is singleton, we may refer to 'probabilities'.

The following two questions are (arguably) the same question:

I have two children; the eldest is a boy. What is the probability that the second child is also a boy?
I toss the fair coin twice; it comes down "H" on the first toss. What is the probability that the second toss is also "H"?

Of course, we are here talking about mathematical children. Whether real children are equivalent to mathematical children is another matter. There are two reasons why they might not be:

Although real children may be tosses of a coin, that coin might not be the fair coin. Worldwide, there seem to be about 105 boy births to every 100 girls, which would mean that gender at birth would be represented by the coin (0.512,0.488).
There is the question of whether or not the gender of a second child is independent of that of the first. Since this is a matter of observation and statistical analysis, the possibility that there might be a connection (and, if so, which way) cannot be completely ruled out. If there is a connection, then -because the data of the first child's gender would be having an effect- we would not be dealing with a probability, so the underlying set could not be singleton; it would, presumably, be (something like) a contraction of S(2) with centre (0.512,0.488).

The situation is much the same as with minted coins. It's just that we don't call them minted children and we do have a significant amount of real-life data.

The whole subject of mathematical children can become very complex, very quickly. For a discussion of some of the points, see the Wikipedia article Boy or Girl Paradox