596 



KNOVV^LEDGE 



[May 12, 1882. 



uljjfobrn tells us that oat of two w'a and three Vs we can make 

 ' , „ .. (lifferont permutations. Now, we havo seen that the 

 chance of any given one of these occurring is 



3 3 7 7 7 3'.7' 



— X — X — X - X _ ; or, -— : 



10 10 10 10 10 10^ 



hence, obviously, the cliance that some one or other of tliese per- 

 mutations occurring is obtained by multiplying their total number 

 into the probability of the occurrence of one out of that number. 

 ^, . . 1.2.3.4.5 3-.7' 



This gives, as the required probability, i 2 x 1 a 3 " ~hF ' "'■''^h 



■1.5 3'.'; 

 inav either be reduced into the fori 



1.2 



viittfn, 



j; (for further roduc- 

 \i . |3 ■ lO- 



its value is 



tion), or may be convenientl; 



3087 

 lOOOO' 



If we notice how this result has been obtaincJ, we rcadil3- deduce 

 the following important law : — If, at each of a set of (n + i.i) trials, 

 there are (p + 5) possible results, all equally likely, p being of one 

 kind and q of another, then the probability that n results will be of 

 the former kind, and m of the latter, is 



!" + "'• » ,„ 

 — p"'i"' ■ 



|ll_|lH '{p + <,)\2^ 



1 give a few illustrations of the application of this law, before 

 proceeding to notice bow the expressions representing these pro- 

 babilities are related to certain well-known algebraical theorems. 



Suppose we wish to determine the probability that in tossing a 

 coin eight times there will be five heads and three tails. Here p is 

 1 and 5 is 1 ; 71 is 5 and m is 3. So that the required probability 

 is — 



1 



that is - 



E 1^ 2« 1.2.,-i 



So that the odds are 25 to 7 against five heads and three tails being 



If we required the odds against five tossings being of one kind 

 and three of the other, without caring whether heads or tails showed 

 oftenest, we must obviously double the above probability, since 

 there must be exactly equal chances for the result five heads and 

 three tails, and for the result three heads and five tails. Thus, we 

 •»et as the chance that five tossings will be of one kind and three of 



7 

 the other th, or the odds 9 to < against such a result. 

 Id ° 



Xow let us inquire what the chance is that the eight tossings will 

 give four heads and four tails. Our formula gives in this case — 

 lj_ 2. 5.C.7.8 i_. or ?i 



|4_ \i ■ 2- ' "''' 1.2.3.1 • 2" ' ' 128 ' 



So that the odds are 03 to 33 against such a result. (The reader 

 >vill readily see why there is no doubling in this case.) 



35 7 7 



Observe that -r^g is greater than ^, but less than jg ; so that 



when a coin is tossed eight times, wc are more likely to get four 

 heads and four tails than either five heads and three tails, or- three 

 heads and five tails ; but we are more likely to get one of these 

 two last results than the first result. 



What, however, is the chance that six heads and two tails will 

 result ? 



Our formula gives 



1 .u .... 7_, 

 61 

 the odds are therefore 57 to 7 against such a result. 



The chance that six tossings will be of one kind and two of the 



other is — 

 32 



It is similarly shown that the chance of seven heads and one tail 



being tossed is —j the chance that seven tossings are of one kind 



and one of another being 



• The symbol | implies tliat all the whole numbers, from one 



up to the number indicated within the symbol, are to be multiplied 

 together. 



The chance that all the tossings give head is 5^ ; the chance 

 1 



'250" 



Wo notice, then, that the most probable number of heads is tour j 

 and in like manner the most probable number of tails is four; bat 

 the most probable assortment of heads and tails is such that there 

 will be five of one kind and three of the other. 



It would follow, therefore, that if two persons of equal fortune 

 were to venture half their fortune on each of eight successive 

 tossings, the most likily of all results is that one or other will bo 

 just ruined at the end of the series of to.ssings. Bat it is etiually 

 likely that one or other will be the loser j and it is rather more 

 likely that they will come off (juits than that one of them (specified 

 beforehand) will be ruined. This supposes that all the eight 

 tossings are completed before accounts are cleared ; and therefore 

 the policy of gambling is somewhat too favourably treated ; for 

 clearly two unfavottrable tossings to begin with, or three unfavom'- 

 able out of the four first tossings, although they might be cancelled 

 by favourable throws if the tossing were continued, would yet 

 complete the ruin of a player, if the money ventured had to be 

 handed over to the winner after each several tossing. 



Let us next take the following example : — 



A die is thrown eight times ; what is the chance that ace is 

 thrown twice (exactly) ? Here the p of our formula is 1, the q is 

 5 (since there are five throws other than ace) ; « is 2 and m is C. 

 Thus the required chance is by otu- formula — 



^= — . --7 ; or 28 • — 



^ |G_ C 6" 



the value of which can be easily obtained either by direct calcula- 

 tion or by means of logarithms. 



If we examine otir formula- 

 |n -l-m 



(P + 0)V" 



We find that it can be aiv«.M into two pnrts. each readily 

 defined. First, there is the expression (p + 'j )"''"■", which obviously 

 corresponds to the total number of possible results when there are 

 {p + Q) possible events at each trial, and (11 -l-m) trials. The other 

 portion 



\n + m 



must, therefore, represent the total number of favourable results, 

 that is, the total number of restilts fulfilling the required con- 

 ditions. It is easily seen that this is so. For in fact, if we take 

 any particular case in which n of the results are of the kind which 

 can happen in 2' ways, and m are of the other kind which can 

 happen in q ways, we see that this particular case can be varied in 

 p" q'" ways. For instance, reverting to our illustrative case, the 

 jiarticular result v: h b h v: may be varied by having any one of the 

 three white balls to give the first 1", by having any one of the seven 

 black balls for the first ■-, any one of the same set for the next f-. 

 and so on ; giving in all 3 times, 7 times, 7 times, 7 times, 3 jk>?- 

 sible variations in which the sequence is ir V h h «■ — that is, 3' 7^ 

 such variations. And the number of possible sequences of n + m 

 results, of which 71 are of one kind and n; of another, is, by a well- 

 known rule, 



In -l-m 



Hence the total number of favourable results is obtained by multi- 

 plying these numbers together, or is. 



In -l-m „ „ 



This expression is the term involving p" q'" iu the expansion of 

 {p + q) to the power (n + m). 



So that our law may be thus expressed. If there are n -t m trials, 

 at each of which some one ei p+ q events, all equally likely, must 

 occur, p of these events being of one kind and q of another ; then 

 the chance that n events will be the former kind and ni of the 

 latter, is represented by the fraction of which the numerator is the 

 term involving p" q'" in the expansion of {p -t- q) to the power 

 (>i +ni), the denominator being the complete expansion. 



This is the chance that there will be eutctlu n results of the 

 former kind. The chance that there will be at least n results of 

 the former kind is obviously obtained by adding together for the 

 numerator all tho terms of the expansion from the first down to the 

 term involving p" q" (both inclusive), the denominator being, as 

 in the former case, the complete expansion. 



