Teaching Statistics
Coincidences often surprise us, suddenly springing up to reveal an unexpected connection between people or things. Sometimes they seem so outlandish as to demand a "supernatural" explanation. Yet anyone familiar with probability theory knows how this notoriously counter-intuitive branch of mathematics can spring big surprises on us. One of the most famous and relevant examples is the so-called Birthday Paradox, which states that in a random gathering of just 23 people, there are 50:50 odds that at least two of those present have the same birthday. Many people find this result very surprising: a recent survey of university students found a median value for the estimated size of gathering needed of 385. (Matthews & Blackmore 1995). So large a gathering is, of course, guaranteed to contain at least one coincident birthday, suggesting that the probabilistic aspects of the paradox evade many people. The same study also showed that people tend to grossly overestimate the size of gathering needed for other types of coincidence.
Part of the explanation for this general lack of insight into the probability of coincidences is that most of us do not go "looking" for coincidences: they "find" us. If instead we made a point of demanding the birthdays of everyone at every gathering we attend, we would soon discover that coincident birthdays are indeed relatively common. We would also get a better understanding of why: firstly, that we are not demanding a coincidence between specific people or specific birthdays, but just any people and any birthday; and secondly, that the key factor is not the number of people at the gathering, N, but the very much larger number of possible pairings of people with which to get a match, N(N-1) /2 (= 253 in the case of N = 23). A more quantitative explanation of the Birthday Paradox can, of course, be given using probability theory. There is, however, no substitute for real-life evidence, and in what follows we outline a simple and appealing demonstration that coincidences really are "out there" - and they follow the predictions of probability theory.
The key source of perplexity with the Birthday Paradox is the low number of people needed to give decent odds of finding at least one coincident birthday. A football match provides an ideal test-bed for this assertion: it has 23 people on the pitch (11 players per side, plus the referee), and it seems reasonable to assume their birthdays are randomly distributed over the year (a point we return to later). If the Birthday Paradox is correct, then in a sample of F fixtures, we expect about 0.5F to contain at least one pair of players sharing the same birthday. However, probability theory allows us to predict several other types of coincidence we should also expect to observe. To see this, we can model the distribution of players' birthdays among the days of the year as a balls-in-urns model, with 23 "balls" being distributed among 365 "urns". A coincidence is then characterised by having at least one urn containing two or more balls, a situation that can be visualised via an "occupancy diagram":
[2] [1,..,1] [0,...,0]
(1) (21) (343)
where the numbers in square brackets show the occupancy of each urn, and the numbers in parentheses represent the number of urns with these levels of occupancy. This diagram represents the case of precisely two of 23 people sharing the same birthday. Calculating the probability of such an arrangement is then a three-stage process: (i) calculation of the number of ways of arranging the urns, U; (ii) calculation of the number of ways of arranging the balls within those urns, B; (iii) multiplying U by B and dividing by the total number of ways of distributing 23 balls among 365 urns, i.e. 365^23. Both (i) and (ii) are given by the standard result that the number of ways of dividing a population of N elements into k sub-groups, of which the first contains r1 elements, the next r2 elements etc. is N! /r1 ! r2 !....r k !. We then obtain the following results:
(1) Probability of at least one coincident birthday
This is 1 - Pr(no coincident birthday), where the lack of a coincident birthday leads to an occupancy diagram of
[1,..,1] [0,...,0]
(23) (342)
U is then 365!/(23!342!), while B is 23!/(1!)23 (0!)342 = 23! and thus Pr(no coincident birthday) = 365-23 x 365!/342! = 0.493, so that Pr(>1 coincident birthday) = 0.507, from which the original Birthday Paradox follows.
(2) Probability of precisely one coincident birthday
The occupancy diagram was given earlier, and leads to U = 365! / (21! 343!), B = 23!/2! and so Prob(1 coincident birthday) = 365^-23 x U x B = 0.363.
(3) Probability of precisely two coincident birthdays
For two pairs of participants to share coincident birthdays, the occupancy diagram is
[2,2] [1,..,1] [0,...,0]
(2) (19) (344)
so that U = 365!/ (2! 19! 344!), B = 23!/(2!)2 and Prob(2 coincident birthdays) = 0.111
(4) Probability of precisely three coincident birthdays
For three pairs of participants to share coincident birthdays, the occupancy diagram is
[2,2,2] [1,..,1] [0,...,0]
(3) (17) (345)
so that U = 365!/ (3! 17! 345!), B = 23!/(2!)3 and Prob(3 coincident birthdays) = 0.018
(5) Probability of one set of triply-coincident birthdays
For three participants to share the same birthday, the occupancy diagram is
[3] [1,..,1] [0,...,0]
(1) (20) (344)
so that U=365!/ (1! 20! 344!), B = 23!/3! and Prob(1 triply-coincident birthday) = 0.007
(6) Probability of birthday on day of fixture
To demonstrate the impact of being specific about the day for which a coincidence is required, we also include the probability that at least one person among N playing on a specific day will be celebrating their birthday. This is 1 - (364/365)^N = 0.061 for N = 23; for Prob(birthday on specific day) to be 0.5 requires N around 256.
Having shown how to calculate probabilities of various types of coincidences occurring in a football fixture, let us now put them to the test.
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ANALYSIS OF FOOTBALL FIXTURES
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To find out if the various birthday coincidences do occur at the rate predicted above, we need a sample of football fixtures, and the dates of birth of all the players and referees. For our sample, we chose the ten Premier Division fixtures played on 19 April 1997, which at kick-off involved a total of 220 players and 10 referees. We obtained the dates of birth of players using Rollin (1996) plus some club data, while the referees' data came from the Football Association (we note that it is not necessary to use referees: the dates of birth of the first substitute played could be used instead). By cross-checking the various dates of birth in all 10 fixtures, we obtained the following results:
Fixture Coincident birthdays (team, date)
Arsenal v. Blackburn No coincidences
Aston Villa v. Tottenham Eliogu (AV; 3.11.72) Yorke (AV; 3.11.71)
Chelsea v. Leicester City Petrescu (C; 22.12.67) and Morris (C;
22.12.78)
Hughes (C; 1.11.63) and Elliott (L; 1.11.68)
Liverpool v. Manchester James (L; 1.8.70) and Wright (L; 1.8.63)
Utd Butt (M; 21.1.75) and P Neville (M; 21.1.77)
Middlesborough v. Johnston (S; 14.12.73) and Waddle (S;
Sunderland 14.12.60)
Newcastle v. Derby No coincidences
Nottingham Forest v. Martyn (Le; 11.8.66) and Halle (Le; 11.8.65)
Leeds
Sheffield Wed v. No coincidences
Wimbledon
Southampton v. Coventry Benali (So; 30.12.68) and Whelan (Co;
30.12.74)
West Ham v. Everton No coincidences
We can now compare these results to the number of coincidences of various types predicted to occur among 10 fixtures using the probabilities calculated in the previous section. The results are as follows:
Type of coincidence Expected Observed No coincidence seen 5 4 At least one coincident birthday 5 6 Exactly one coincident birthday 4 4 Exactly two coincident birthdays 1 2 Exactly three coincident 0 0 birthdays Exactly one triply-coincident 0 0 birthday >1 participant with birthday on 0 - 1* 0 19.4.97
Table 2: Comparison of predicted and observed number of coincidences in 10 fixtures
Table 2 shows impressive agreement between the predictions of probability theory and the observed number of coincidences. In particular, it confirms the theoretical prediction that the less specific a coincidence is, the more likely it is to occur: getting any two players to share some birthday proved possible in four out of the 10 fixtures, but not one of all 230 participants had a birthday on the specific day of the match.
As we have seen, coincidences tend to be considerably more likely than we might think. They become more likely still if we allow a little latitude into our definition of what constitutes a coincidence - for example allowing birthdays separated by no more than r days of each other to constitute a "hit". As before, we can model this "near-miss effect" using a balls-in-urns model as before; the argument is somewhat more involved (see, e.g. Naus 1968), and leads to
Pr( > 2 birthdays separated by < r days) = 1 - [(364 - rN)!3651 - N/(365 - (r + 1)N)!]
Thus, for exactly coincident birthdays we have r = 0, while for birthdays either on the same day or on adjacent days we have r = 1. To compare these theoretical values with the reality of our football matches, we set N = 23, leading to a near-miss probability of 0.888 for r = 1: that is, we expect about 9 of the 10 fixtures to feature participants whose birthdays are within a day of each other. In fact, all 10 of the matches have at least two players with birthdays within a day of each other - again, impressive agreement with the predictions of probability theory.
The ability of the "near-miss effect" to boost considerably the chances of coincidences can be seen even at the level of each team. Setting N = 11, we find that the probability of individual teams having at least two birthdays separated by no more than 0, 1, 2, 3, 4 and 9 days are 0.141, 0.371, 0.543, 0.672, 0.767 and 0.948 respectively. Table 3 compares this with observation:
Type of coincidence Expected Observed At least two coincident birthdays 3 6 At least two birthdays < 1 day 7 13 apart At least two birthdays < 2 day 11 17 apart At least two birthdays < 3 day 13 18 apart At least two birthdays < 4 day 15 18 apart At least two birthdays < 9 day 19 20 apart >1 participant birthday on 1 1 (18-20)/4/97
Table 3: Comparison of expected and observed number
of "near-miss" coincidences among 20 teams
Once again, the overall agreement between theory and observation is impressive: as before, we see that as the "window of opportunity" given to the near-miss effect is widened, the number of coincidences increases. It is worth noting that the biggest increase comes from allowing birthdays that fall on adjacent days also to count as "hits": this small concession doubles the number of coincidences. It is also worth pointing out that - as with our earlier comparisons of theory with reality - the deviations between the two tends to favour the existence of more coincidences. This is a reflection of the fact that there is a significant preponderance of players' birthdays in November and December, and deviations away from a uniform distribution of birthdays always tend to boost still further the number of observed coincidences.
We have shown that football fixtures provide a simple and convenient way of investigating the prevalence of coincidences. The raw data is of a familiar type, is easy to obtain from published sources, and motivates the use of simple combinatorics in making predictions about what should be observed. Our own previous research suggests many people will be very surprised by the results.
Acknowledgements
RM thanks Simon Singh for inspiring him to use football matches as a way of probing coincidences, John Haigh and an anonymous referee for very helpful comments, and Ms J A Stearn of the Football Association for the dates of birth of referees.
References
Matthews, R.A.J., Blackmore, S.J. 1995 Why are coincidences so impressive ? Perceptual and Motor Skills 80 1121-1122
Naus, J. I. 1968 An extension of the Birthday Problem The American Statistician 22 27-29
Rollin, J. 1996 Guinness Soccer Who's Who 10th Edition (Guinness Publishing, Enfield) .