5tO 



KNONA^LKDGE 



[April 28, 1882. 



ani^urr^ to Corrc^pontirntsf. 



* ^* All i-ommttnie<ifiong for the Editor requiring early attention thould reach the 

 Office on or be/ore the Saturday prrcedintj the current ienue of Enowlbdob, the 

 iitcreaeing circulation of which compete ue to go to preee early in the tceek. 



Hints to Corrkspondrnts. — 1. No que»tione asking for tcientijie information 

 can be anriarred through the poet. 'I. Lettere gent to the Editor for corrt-epondente 

 cannot be forvarded ; nor can the names or addressee qf correspondents be given in 

 anjwer to private inquiries, 3, Correspondents should write on one side only of 

 the paper, and put drawings on a separate lenf. 4. Each letter should have a title, 

 and in replying to a letter, r^erence should be made to its number, the page on 

 which it appears^ and its title. 



3. Ralph. — Vour problem is not definite. An infinite number of 

 ellipses may be drawn toucbiug two straight lines at two given 

 points in the same. — H. C. SrANnAGE. Table of pigments quite 

 unsuited to us. — F. C. M. The mean temperatures in Whitaker 

 are as it were smoothed down, so as to get rid of such peculiarities. 

 But in the mean curve for the last scvonty-five years the cold days 

 of April and other three cold days in May are clearly seen. — 

 S. Barber. Could not say till article received. H. Weatherhead. 

 American Exchange, MO, Strand. — Senex. No, we cannot answer. 

 J. Frasee. Your rea.soning quite unsound, but it is impossible to 

 explain the matter fully here. The rotation of earth on her 

 axis has nothing to do with the theory. — D.nommah. Awfully para- 

 doxical, as you say. — J. Ci.ift. Thanks on Mr. Muybridge's behalf ; 

 but he has already photographed swiftly-running hounds. — T. E. 

 Allinson. Letter received, and shall appear. — M. B. Alder. Gravity 

 very essential to Dr. Siemens' theory. More about it hereafter. 

 The change of earth's axial pose due to precession cannot be dealt 

 with briefly : must be very fully illustrated. — A Hater of Super- 

 stition considers any reference to " The Almighty," " His glory," 

 " God's ways," and so on, unsuited to a scientific journal. His 

 objections, however, apply only to dogmatic, not to natural re- 

 ligion, which can never be out of place in dealing vrith science. — 

 F. Miles. For astronomical observation generally, not for observ- 

 ing stars by day. — Antarctic. The low barometer of Antarctic 

 regions was what suggested the theory. — J. Ralph. Cannot give 

 further space to vegetarianism. — Curiosity. Letter received, and 

 to appear when space permits. — J. McDowell. Tlianks. — A. H. 

 Empson. You would do well to accept Messrs. DoUond's view of 

 the matter ; 80 is a liigh power for a 2J inch object-glass. — H. C. 

 Wtatt. We have added Lyons. — .T. A. Ollard. A room may be cut 

 off from the rest, by suitable devices. — C. Staxiland Wake. I 

 should be willing to lend the pyramid blocks ; but the articles are 

 shortly to be published in book form. You will see that in the con- 

 cluding paper of the article the religious use of the pyramid is 

 referred to. — Geo. Miller. — The stern aspect of the hogs in the 

 Scottish rhymes is amusing ; bat lectaribus nosiris major debctur 

 reverentia. — G.Shopland. Fear Newcomb's book is dear, about 18s., 

 I think. — W. Smith. Quite unable to answer letters by post. Do 

 not know who is the editor of the Scientific America}}. It is pub- 

 lished weekly by Munn & Co., 261, Broadway, New York. Pro- 

 bably the reports of the Meteorological Society would suit you. It 

 would be invidious to say which we think the best text-book of 

 meteorology, and contrary to our rules. — F. Gibes. The process is 

 called fermentation, and is now recognised as depending on the 

 development of Uving germs. The subject is one for an article, 

 and we hope soon to find room for one. — J. Muhrat. I have 

 endeavoured to show that, in my opinion, the P.D. system is 

 utterly untenable. — A. Bexett. You ask us how you can make 

 Balmain's Patent Luminous Paint for your own use ? Is the 

 question quite fair ? What are patents intended for ? — 

 Herbert J. L.\mb. The atmosphere so forming round the moon 

 would be of very small density. — John Hamek. Since noted. — 

 Young Astronomer. The subject shall be treated soon. Thanks 

 for encouraging letter. The slia])e of a halo depends in no way on 

 shape of luminous body. — W. P. Hamsby. We referred to England. 

 There is no evideuce of any great perturbation in this country, 

 A.I). 1,000.— Sandford Flem'ing. Thanks; but "Time Reform" 

 rather a dry subject for readers of Knowledge. — W. H. Shirley. 

 " Nautical Almanac " for 1882, 1883, &c., of Messrs. Mm-ray, price 

 2s. Gd. — N. A very neat " martingale " will discuss it presently; 

 but, in passing, note that as for every win you score out two 

 figures, and for every loss you add one, you cannot win one 

 "revolution" until the number of wins exceeds the number of 

 losses by the number of fig\ires first set out. The assumption that 

 this must happen before yi)ur pockets are cleared out is the 

 " fallacy." — Bkccablnga, J. Rkiu, Erix-go-bragh, Joseph Wallis, 

 Beccabunga, J. Wilson, C. J. Caswell, Photographer, R. F. S., 

 Antarctic, Julias, E. W. Hokton, H. D., J. F. R., H. R. L., M. 

 Uaugreave, P. K. Pattinson, L. T. P. K. Letters forwarded to 

 contributor on snch subjects. 



Sartor Rcsartus, J. S. P., Thomas Bliindem, W. B., Mat, S. C. 

 Wood, J. R. Musgrave, T. Y. S., G. Redfcrn, K. P. Wallis, M. 

 Shortrede, P. Q., L. T. Turvey, J. C. B., queries already answered, 

 too vague, or otherwise unsuitable ; P. Jackson, S. Homo (thanks), 

 jr. .\., Professor of Mathematics, Ch. Harrison, Pertinai, Sic 

 tran.iit. L. T. R., Providence, M. Purvis, Exccllens, &c. 



(J^ur iHatbcmatiral Column. 



PROBABILITIES. 

 By the Editor. 



LET us next consider cases not altogether so simple. I propose 

 now to establish what may be regarded as the fundamental 

 proposition of direct probabilities. To introduce it, I take the fol- 

 lowing simple illustrative case : — 



Suppose that in an urn there are three white balls and seven 

 black ; and in another urn two white balls and three black, what i$ 

 the chance that when aball is drawn from each urn, both the drawn 

 balls ivill be white ? 



Applying to this problem the two fundamental principles of the 

 science of probabilities, we inquire, fir.st, how many possible events 

 there are, and, secondly, how many are favourable. Now any one 

 of the ten balls in the first urn may be drawn, and any one of the 

 five balls in the second urn. So that any one of the first set of ten 

 balls may appear in company with any one of the second set of five. 

 There are thus 50 (10 times 5) possible events. Again, the pair of 

 drawings may result in giving two white balls in 3 times 2 different 

 ways ; since any one of the tliree white balls in the first am, by 

 being drawn in company with any one of the two white balls 

 in the second urn, would give the required result. .Six events, 

 then, out of fifty are favourable ; and, therefore, since any one of 

 the fifty events is as Ukely to occur as any other, the chance of 



drawing two white balls is — or 



^ 50 25 



We can see from the method here applied to a special case that 

 the following general nile may be deduced : — If there are two inde- 

 pendent events, and the first can happen in p ways out of p -r 5 all 

 equally likely, and some one of which must happen, while the 

 second can happen in p' ways not of p' + q' all equally likely, and 

 some one of which «i!(s4 happen; then the chance that both events 



will happen is 



— , that is, 



or the 



(p-i-g) (p' + g') ' V + 1 P' + 9' 



chance of both events happening is obtained by multiplying to- 

 gether the chance of each considered separately. 



It follows obviously that if we add a third independent event, 

 which may happen in p" ways out of p' + q" all equally bkely, and 

 some one of which must happen, then the chance that all three 

 events will happen is 



V P P" 



(p + q) ip' + q) (p" + 9") 

 or the product of the three several chances. For we have already 

 seen that the chance of the two first events happening is the pro- 

 duct of their several chances. We may regard this as a single 

 chance. Taking, then, the thu-d event, we have the chance of its 

 occurring as well as both the former, equal to the product of the 

 chance of the third event by the chance of both the former hap- 

 pening ; that is „-i- — i, X >— / T- JT 



^ ^' p" + q" (p + 3) (P + 9)- 



And so we obtain this general law, that the chance of several 

 independent events all happenimj is equal to the product of the chances 

 of the several events. 



As examples of the application of this rule take the following : — 



Ex. 1. — Tlie chance that a horse ivill icin a certain race is reckoned 



at L^ or the bettimj is even upon him; the chance that an oarsman 



u'ill win a certain boat-race is reckoned at - (the odds 2 (0 1 against 



3 

 him) ; the chance that a county v;ill u-in a certain cricket match 



against another county is reckoned at — (the odds 3 (0 1 against tht 

 former). What are the odds against all three evods happening ? 



The chance that all three events will take place is - x _ x _, or 



2 3 4 



— . Therefore, the odds against all three happening ai'c 23 to 1. 



