Volume XIII 
OCTOBER, 1921 
No. 4 
ON RANDOM OCCURRENCES IN SPACE AND TIME, 
WHEN FOLLOWED BY A CLOSED INTERVAL. 
By G. MORANT, B.Sc. 
(Being a paper read to the Society of Mathematical Statisticia 
and Biometricians, 1921.) 
I. Introductory. 
The object of the present paper is to investigate a little more fully than has 
hitherto been done the formulae for random occurrences in space and time, and to 
test their adequacy on the basis of experimental data. 
The fundamental formula, which was first stated and proved by Whitworth*, 
is obtained as follows: An event happens at random once in a period rn, therefore 
ht 
its chance of occurring in an interval of time or space is — , and of its failing to 
° ^ m ° 
occur ^1 — — j . If we now take n such intervals, its chance of not occurring in 
time nht is (l j , accordingly if we make t = n8t and suppose n to become 
indefinitely large and St indefinitely small, we have 
t 
Lt fl-— y' = e"'«, 
,i=ac\ n'/nJ 
which is therefore the chance of non-occurrence during an interval t, when the 
occurrence measured on a very long period averages once in the interval m. 
Whitworth gives no experimental data to confirm this theory and apparently 
did not fully recognise its immense importance, if confirmed, for medical and socio- 
logical " events." That is to say, as a means of distinguishing between random and 
associated occurrences, for which it offers, with some expansions, a most valuable 
criterion. 
At first sight it might seem an easy task to justify or refute s.uch a law of 
distribution by actual experiment. But in endeavouring to find how such a series of 
events may be obtained by observation or experiment there are serious difficulties 
to be encountered. Dr M. Greenwood suggested dealing experimentally with the 
matter, and he drew numbered counters from a bag and considered the differences 
between successive numbers to be intervals between events. But such a case does 
not fully accord with theory, for the counters being numbered by units the intervals 
will be clustered in masses differing by a unit. The chief difficulty of any experi- 
mental demonstration lies in allowing for a large number of simultaneous occur- 
rences. For example, if we take taxi-cabs passing a given lamp-post in a given 
street in one direction, it is perhaps only possible for one or two to pass at one 
* Choice and Chance, Cambridge, 1901, 5th Edition, p. 200. 
Biometrika xiii 20 
