P. V. SUKHATME AND V. G.
PANSE
77
catch infection is ¼ × ¼ × ¼, since the event can happen in only one way, and
equals 1/64 . Tabulating we have—
Event observed
All three animals free from infection
....
Alternative events—
(i) Two animals free from infection and third not
(ii) One animal free from infection and the other two not
(iii) None free from infection
......
Probability
27/64 = 0.4219
. 27/64 = 0.4219
. 9/64 =0.1406
1/64 =0.0156
Total probability is 1. The table shows that in experiments with three
animals we should expect the event observed to occur 42 times in 100 experi-
ments even if the treatment were valueless. In an experiment with three ani-
mals therefore the probability of none catching infection is large and again we
shall not be justified in attaching a real value to the treatment.
Suppose we increase the size of the experiment to four animals and observe
that none caught infection, whereas normally one out of four would be expected
to be infected. Can the departure of the observed result from the expected ratio
be considered large enough to have arisen from the effect of the treatment
and not from chance causes ? As in previous example to answer this question
we shall need to know the probability that a result as good as the one observed
or better giving a deviation as large as 1 or larger would occur through chance
causes. In a sample of four only one such event, viz. all four animals remaining
free from infection is possible and the required probability is therefore (¾ ) 4 =
0.3165. This means that the result such as the one observed (and better)
would arise through pure chance nearly 32 times in 100 experiments with four
animals. Obviously in a single experiment with four animals a result better
than past experience is quite a likely event to happen in the absence of real
treatment effect and we cannot deduce that the departure of the result ob-
served from the expected is significant of the treatment effect.
In experiments with a larger number of animals we can similarly calculate
the probabilities of the observed and larger departures arising by chance and
thus make what is commonly known as the test of significance of the departure
of the observed result from the expectation. Let us suppose that in an ex-
periment with 12 animals one was found to catch infection in place of the
expected number of three. Can the departure of two of the result observed from
the expected number be considered to be sufficiently large to indicate that the
treatment is of value ? The answer would depend on the value of the pro-
bability that a departure as large as 2 or larger would arise even if no treat-
ment was given. The probability required is obviously the sum of the
probabilities that only one out of 12 animals catches infection and none catches
infection. This will be 12 × ¼(¾) 11 + (¾) 12 = 0.1586 or 16 per cent. The value
of the probability is smaller than any we have worked out above but is not so
negligible that we may ascribe the result observed with fair confidence to the
effect of the serum treatment.
With 12 animals and none catching infection, the interpretation will,
however, be different since the departure from the expectation in each clase
' infected ', and ' non-infected ' is now 3 and the probability of this departure