342 



NA TURE 



[March i6, 1922 



With an unbiassed and experienced forecaster it may 

 be assumed that in the long run " Yes " occurs among 

 the forecasts about as frequently as the event occurs. 

 In a large number n of trials, " Yes " will therefore 

 be forecasted on Pw occasions and the expenditure on 

 precautions will be a . Pw. 



The event will occur on P« occasions of which f . Vn 

 will have been forecasted. The remaining unfore- 

 casted occasions will number Vn{\-p) and will entail 

 an expenditure of 6 . Pw(i - />) by damage. The total 

 expenditure will therefore be 



a.P« + &.P«(i-^). . . (I.) 

 The cost of the forecasts is assumed to be negligible in 

 comparison with a and h. 



If precautions are always taken the expenditure 

 will be 



a.n (II.) 



If precautions are never taken the expenditure will be 

 h.Vn (III.) 



We have now to compare the amounts involved by 

 I., II., and III. and see which is the least. 



I. may be written b .Fn- {b .Fn .p-a . Fn). 



The condition for I. to be more economical than 

 III. is therefore 



b . Fnp -a.Fn>o 

 or p>ajb. 



That is to say, the probability of a correct forecast 

 must be greater than the ratio of precautionary ex- 

 penditure to possible damage. Unless forecasts are 

 very bad or precautions very expensive this condition 

 will be very easily fulfilled. 



The condition for I. to be more economical than 

 II. is 



b.Fn{i-p)<a.n{i-F), 

 which expresses the fact that the loss due to possible 

 damage must be less than the saving through omission 

 of precautions. This may be written 



i-p<{i/F-i).alb. . . . (IV.) 

 With given values of P, a, and b, IV. sets a limit to 

 the allowable error in forecasting. For example, 

 suppose the chances of the event are even, that is 

 P = o-5, the following results are obtained for the 

 limiting values : 



a/b = o-oi 0-05 o-io 0-20 

 p = o-gg 0-95 0-90 o-8o 



If the probability of a correct forecast is not above 

 0*90 it is therefore disadvantageous to base pre- 

 cautionary action on the forecasts unless it costs 

 more than il. to save 10/. worth of damage. 



With lower probability of the event things become 

 much more favourable for ( i ) . Thus suppose P = o • 25 

 (chances 3 to i against the event) we obtain the follow- 

 ing results : 



a/i = o-oi 0-05 o-io 0-20 

 p = o-gy 0-85 0-70 0-40 

 If in this case the probability of a correct forecast is 

 0-90 and alb = o-io, the expenditure involved per 100 

 trials, il. being the precautionary expenditure, works 

 out as follows : 



Taking no precautions . . . 250/. 



Taking precautions on all occasions . 100/. 



Taking precautions only when the event 



is forecasted 30/. 



In practice, the forecaster, being aware that the 

 object of the forecast is to avoid loss by damage, 

 would be biassed in favour of forecasting the event 

 and would only forecast " No " when the odds were 

 considerably against an occurrence. Such a bias 

 would tend to increase p, which was defined as the 

 probabiUty that an occurrence of the event would 

 be successfully forecasted. Against this we would 

 have to set an increased expenditure through needless 



NO. 2733, VOL. 109] 



precautions, and if the forecaster were too cautious 

 this might outweigh the gain resulting from a decrease 

 in unforecasted occurrences. A moderate degree of 

 bias would, however, obviously result in a gain in 

 most cases. 



In practice, therefore, the financial results of utilis- 

 ing weather forecasts are likely to be even better 

 than those calculated above. 



There is reason to believe that the commercial 

 possibilities of weather forecasts are not fully appreci- 

 ated. These calculations will serve to show that 

 considerable gain may result from their utilisation, 

 allowing a reasonable margin of error in the forecasts. 



E. G. BiLHAM. 



Age Incidence of Influenza. 



The explanation given by Miss .'\. D. Betts 

 (Nature, February 23, p. 240) has always seemed 

 to me an obvious and adequate one. The reply of 

 " The Writer of the Article," that follows it, is 

 admirably clear, and therefore helpful ; but I think it 

 really supports the explanation it ostensibly opposes. 



Influenza is clearly a disease the incidence and 

 severity of which depend more on the man than 

 on the germ. Some people (apart from the intensive 

 infection and exhaustion of overcrowding and nursing) 

 seem practically immune to it. But all such im- 

 munity is probably partial and relative. The 

 successive milder attacks, sometimes not called 

 influenza, that follow a well-marked attack show 

 that the patient is not immune ; and any relative 

 immunity is probably largely due to these later 

 successive attacks being treated in time. 



In the same way the degree of immunity induced 

 by preventive inoculation is, like other immunity, 

 partial and relative. The inoculated may hope, at 

 least for a time, for substantially increased resistance 

 to infection (but not absolute immunity no matter 

 how intense tlae infection) , and for slighter and briefer 

 attacks if they catch it. 



Further, all such relative immunity, if originally 

 present, tends to be destroyed and the severity of 

 the disease and its associated complications increased 

 by exhaustion, whether due to virtuous and patriotic 

 overwork or to vicious dissipation. I do not there- 

 fore oppose point (4) of " The Writer of the Article," 

 which is not, however, in dispute here. 



But as regards his other three points — 



(i) Under end-of-war conditions our young people 

 with unselfish patriotism worked to exhaustion, 

 and largely under conditions involving intensive 

 infection ; and as a secondary factor had inadequate 

 food in health and when ill. Again, working as 

 they did largely away from home, the dangerous 

 work of nursing them fell largely on those of their 

 own age, and not, as in normal times, on their 

 parents and elders. 



The latter, on the other hand, were not only 

 spared the exhaustion and intensive infection of 

 nursing them, but the " rationing," so trying for 

 the young, was good for us older people, to whom 

 habitual over-eating and drinking is apt to be one 

 of the chief causes of exhaustion and consequent 

 liability to disease, at an age when we should be 

 most immune. 



Doubtless the aged were hard hit by the sorrows 

 of the war, which in the end shortened the lives of 

 many ; but I take it that, except in the countries 

 actually invaded and ravaged by the enemy, war 

 conditions did not make them more liable to influenza. 



(2) Observations on influenza in boarding schools 

 show that those boys who do not readily catch it, 

 and, under normal home conditions, would escape prob- 

 ably entirely, if exposed long enough to the intensive 

 infection of crowded rooms at school, succumb in the 



