614 



♦ KNOWLEDGE ♦ 



[May 19, 1882. 



<!Pbitiiaii). 



WE rpjjrot til have to nnnonnce the death, at the early ngc of 

 tliirty-two, of a promisinf,' young science tcMcher and 

 lecturer, Mr. Thomas Dnnnian, Lecturer on Physiology at the Birk- 

 bock Institution, and Physical Science Lecturer at the Working 

 Men's Collcjje. Ilis early education was limited, bat his reading 

 was wide and his memory remarkably retentive. It was always in 

 spite of his surroundings that he went on adding to his stock of 

 knowledge. About seven years ago ho took charge of the Physio- 

 logy class at the Working Men's College in Great Ormond-strcet, 

 where liis class was one of the largest and most popular in the 

 College. The practical results shown by the examinations at South 

 Kensington attested the thoroughness of his teaching. Like snccess 

 attended him in other courses of lectures in other branches of 

 science. At the Birkbeck Institution, where he sncceeded Dr. 

 Aveling as Physiology Lecturer, his work was much appreciated. 



In 1879 he published a very useful glossary of " Biological, 

 Anatomical, and Physiological Terms," and four of his lectures had 

 appeared, " The Mechanism of Sensation," " The Starlit Sky," 

 " Prehistoric Man, ' and " Volcanoes and Coral Reefs." lie 

 contributed to Cassell's " Science for All," to Ward & Lock's 

 " Universal Instructor," " Amateur Work," and several other 

 publications. 



Ho married early, and the effort to support his family by science 

 teaching and lecturing may fairly be said to have cost him his life. 

 During the past two years there were warnings that his energies 

 were being too strongly t.axed, but they were unheeded, and at the 

 beginning of the present year brain troubles became markedly 

 apparent, and he was obliged to give up work ; but it was too lato. 

 He gradually grew worse, and died on the 9th .inst., leaving a 

 widow and two children, for whom he had been unable to make any 

 provision. 



0\\v iilatJ)fmaticaI Column. 



THE LAWS OF PEOBABILITY. 

 By the Editor. 



AS an illustration of the rules established in our last, take the 

 following : — 



There are in a bag three white halls a)id sex-en black balls; a ball 

 is drawn at ranAitn and replaced, and this process is repeated five 

 times ; what is the probability that at least two white balls will be 

 draicn ? 



Applying the rule, we must suppose 3 + 7 expanded by the 

 binomial theorem to the power 5, the complete expansion being 

 thus written : — 



3' + 5.3'.7 + 10.3'.7' + 10.3».7' + 5.3.7" + 7= 

 then the fraction obtained by writing the first four terms of this 

 expansion over the whole expansion represents the chance than at 

 least 2 white balls will be drawn. 'The whole expansion is, of 

 course, equal to 10^ or 100,000 ; and the sum of the first four terms 

 is easily found to be 47,178, so that the required probability is 

 47178 

 100000' °'" "*'»'''? one-half. 



It need hardly be remarked, however, that the practioil applica- 

 tion of this rule is not always quite so easy as in the above instance. 

 Tables have been constructed for the detei-mination of approximate 

 values when n + m is large and direct calculation out of the 

 question. 



Of course, the chance that at least two black balls will be 

 drawn is given by taking the last four terms of the expansion for 

 numerator. In this case the calculation is even easier than in the 

 former, though it would be less easy if the student proceeded 

 directly to calculate the value of the four terms, and then to add 

 them together. There is no occasion for this, however, for he 

 knows that the total expansion of 10 to the power 5 is 100,000, and 

 he has only to deduct from this the sura of the first two terms — 

 that is, 3,078, leaving 96922. The required probability is therefore 

 96922 ... 1 1- J 



lOOuOU ' "'' '"°''*' " twice as great as that of drawing at least 

 two white balls. 



It is easily seen that by precisely such reasoning as we have used 

 to establish the law discussed above, we can obtain the following 

 law : — 



If at each trial there are p + q + r possible results, all equally 

 likely to occur, of which p are of one kind, q of a second, and r o"f 

 a third, then the chance that in (n + m + I) trials n are of the first 



kind, m of the second, and I of tho third, is represented by the 

 fraction — 



\n + m + l p' q" r' 



Here, too, as in the former case, the expression for the probability 

 is divisible into two parts : a denominator, the expansion oi (jp+q + r} 

 to the power (n + m + 1) ; and a numerator, the term of this expansion 

 involving p" q" »'. And if we require the jirobabiiity that at least 

 n of the results will be of the first kind, and at least m of the second, 

 we must for a numerator add together all those terms in the expansion 

 of (p-K/ + r) to the power (n-nn-l-J) which involve p", p''*^', p'*', 

 &c., and ulso q", q'"*' (;"+', ic, that is, all terms in which the power 

 of p is not lower than ti, and the power of q not lower than m ; so 

 if results of the first and third, or of the second and third kind are 

 in question. Of course, if wo only require to know what is tho pro- 

 bability that n, at least, of the results will be of the first kind, the 

 problem belongs to the former case. 



The extension of these considerations to cases where there are 

 four possible classes of result, or five, or more, will be a simple 

 matter to the algebraist. The following example will be more in- 

 teresting to the general reader than a mere statement of the lawj 

 Ijut it will be well to notice that the formula for all such cases bears 

 precisely the same relation to that last given that this formula boars 

 to the former.* 



The letters forming the word " Mississippi " are marlced on eleven 

 tablets, all similarg shaped, and placed in n bag. A letter is drawn 

 f om thu bag at randnm and replaced ; and tins is repeat- d twenty- 

 thrcH times; what is the probability that these twenty-three diawings 

 will give 3 m's, 8 i's, 7 s's, and 5 p's. 



The bag contains 1 m, 4 i's, 4 .<f's, and 2 p's, or eleven letters 

 all, and the required probability is — 



|23 VA'A'.2' 



|3 |8~|7 i5 ■ 11" 

 which mav be written — 



|23 2» 



\l~\S_ |7_[5'll^ 

 and is readily calculable by logarithms. 



The value of the probability in this and all similar cases is not 

 changed when the number of possible results of each kind are mul- 

 tijjlied in the same proportion. Thus, if the bag contained 20 m's, 

 hO i's, 80 s's, and 40 p's, we should obtain the same value for the 

 probability above required, as in the case actually described. 



When the number of possible results of each kind is very great 

 indeed compared with the number of trials, we get appreciably the 

 same probability whether after each trial matters are restored to 

 their condition before the trial or not. Thus if a bag contain a 

 million red balls, a million white, a million black, and a million 

 green balls, we should get the same probability for the result of 

 twelve drawings (say) whether after each drawing the drawn ball 

 were replaced or not. The difference, at least, is not appreciable. 

 Hence we get the same probability as respects a single trial in 

 which twelve balls are drawn at once, as for twelve several draw- 

 ings (followed by replacement). 



Next week we propose to give the solutions of several problems 

 which have been standing over for some time. Our papers on 

 " Probabilities " will probably be concluded in the two first numbers 

 of Vol. II. 



(JPur C&fSs Column. 



THE INTERNATIONAL VIENNA TOURNAMENT. 



[By Telegram.'] Cafe Reichsrathpark, Vienna, 



Tuesday night. 

 The following is the score of the English players : — 



Mason 5J I Blackburn 4 



Mackenzie 5 Steinitz 2^ 



Zukertort 4i | Bird i 



We have grave apprehensions that Steinitz's health must have 

 broken down. On Friday, when playing .with Captain Mackenzie, 

 he overlooked that he could win a piece on his twenty-second move 

 — the consequence being that the game was drawn. On Saturday 

 he fared still worse ; he lost to Zukertort. His score then stood at 

 2.J. On Monday he had to play Kruby, and on Tuesday Ware. As 



* At p. 3t9, No. 16, will be found a convenient formula for the 

 expansion of a multinomial to a positive integral power. 



