MATHEMATICAL SECTION. ro 
e = the number of coincidences or verifications, 
} = the inference-ratio, or that part of the success which is due to 
skill and not to chance, and which may be called the degree 
of logical connection between event and prediction. 
Since success is proportional to each of the two fractions 
c c 
—~(gnd'—s 
0 p 
it may be represented by their product 
é 
op 
ak ; 
The fraction {> Tepresents the ratio of random success, and 
0 : : awh e ¢ 
therefore = verifications out of p predictions are to be ascribed 
to chance and must be subtracted throughout. The remainders, 
0 0 
0 — ‘and p— i, 
8 8 
represent fields which chance leaves for science to conquer; and 
PE ol 
represents the portion of each which science does conquer. Hence 
he We 
fe ee 
7B DP Opa —oya— py 
8 8 
By another method, 
< = the probability that any single occurrence will be predicted 
in some manner. 
P —: 
8s— 
c bee , 
9 = the probability that any single date of non-occurrence 
will correspond to an unsuccessful prediction = the general 
probability of unskillful prediction in any case. 
Subtract from the probability that any single occurrence will be 
predicted in some manner the general probability of unskillful 
prediction, and we have 
= _ e = = the probability that any given occurrence will be 
skillfully predicted. 
