540 



KNOWLEDGE ♦ 



[Afbil 21, 1882. 



flotrd on 2vt nnti ^rirnrr. 



An czporinicnt with a (yntcm fi>r iininf; pctrolunm inataad of coni 

 for fui'l wim tricil on llio l>onK Iftlnnd Knilroa)) rocently, oncl was 

 proiKiunced n Hticccaii, The tniii) was run on sclicdalo time, and tliu 

 co«t wim $1.'J0, ft« i'OMi|mro<l with $2, thf> price for coal. The new 

 fad is a va|x>iM' produeed liy the intermingling of jetii of petroleum, 

 iin|M'rheat«d Blcam, and hot air. 



A New and intorentin^ proof that the earth is round has been 

 prei"rntc<l hy M. Pnfonrin a paper recently read before tho Helvetic 

 Society of Natural Sciences. In calm weather tho imn^cs of dihlnnt 

 objeels reflected in tho Lake of Geneva showed just exactly the 

 same degree of distortion which calculation would predict through 

 taking into consideration the fignrc of the earth. 



UriKNT investigations by Dr. llann indicate that tho mean tem- 

 peraturo of tho southern hemis[dierc is tho same as that of the 

 northern, but between 10 degrees and 15 degrees south latitude, tho 

 8o<ithem hemisphere becomes warmer than the northern in tho same 

 latitude, and this diCferenco continues at least to the confines of the 

 hypothetical Antarctic continent. 



Gray's well-known work on anatomy has been translated into 

 Chinese, and the translator, tho late Dr. Osgood, is said to have suc- 

 ceeded in giving Chinese names to the multifarious and minute 

 structures which constitute the human body — a difficult task, as the 

 Chinese know scarcely anything of anatomy, or of the functions of 

 the various organs of the body. 



A Geobgia.m, of scientific attainments, residing at Darien has dis- 

 covered that len-scs for telescopes can be manufactured from the 

 virgin drip of rosin. The largest lens made of glass is only thirty 

 inches in diameter. This magnitude can be greatly increased by 

 the new method, and consequently there is no telling what 

 wonderful astronomical results may (low from its adoption. Gentle- 

 men who aie conversant with science say that tho Darien discovery 

 is worthy of a thorough test. 



A NEW plan to deaden floors has been patented, and is being 

 tested in a new building at Philadelphia. A si.\-by-threo plank is 

 inserted between each joist two inches from the bottom of the 

 joists, and projecting four inches beneath. Underneath the inter- 

 vening planks the ceiling boards are nailed and the space filled 

 with sawdust to within an inch of the joists. By this method the 

 waves of sound are carried off, and it is claimed that tho most 

 vigorous hammering cannot be heard in the story beneath. 



©ur i^atlKmatiral Column. 



THE LAWS OF PROBABILITY. 

 By thb Editor. 



AT first sight nothing seems clearer than that the answer given 

 by ninthematieians to tho Petersburg problem is untrue. I 

 have even heard persons to whom the problem and its answer have 

 been submitted assert that no amount of reasoning would convince 

 them that so preposterous a solution was just. Unfortunately, the 

 reasoning given in treatises on probability, through sound, is 

 commonly too recondite to convince these sceptics. Let me repeat 

 the jiroblem, and ro-state the answer; and then let us ti^ to see our 

 way to a clear interpretation of tho seeming paradox. The problem 

 runs thus : — 



Each person in a certain lotterj- is to stake jEx on the following 

 conditions :— A coin is to be tossed until head appears ; if head 

 comes at the first toss tho person is to receive £2; it at the 

 second toss, he is to receive £1; if at the third, ho is to receive 

 £8 ; if at the fourth £1G, and so on. Kequired the value of j. 



The startling answer is that x is equal to infinity ; in other words, 

 that though each person staked a sum never so great, the " bank " 

 would lose. 



Xow it seems so obvious that if a large sum were paid for a 

 rhanco in tho lottery, the speculator would lose, that it is difficult 

 to bolicvo that some fallacy does not underlie tho reasoning by 

 which the above answer is obtained. Accordingly oven first-rate 

 niathematicmns (like d'Alembert) have questioned the justice of 

 thenniwer. "iet I believe I shall be able to convinco even non- 

 roothcmaticjans that the answer is sound. 



Tho main objection ia founded on the difficulty of believing that 

 in any series of tnals, howcrcr long the series might be, tail would 



be tossed many times running. For example, a sequence of twelve 

 tailii hoemg utterly unlikely to occur even in many millions of trials. 

 Kspocially does this seem to bo the case, when wo try to consider 

 the case of a person who should keep on continually tossing a coin 

 until he had tossed twelve tails in succession. Ue might toss twenty, 

 thirty, a hundred, nay a thousand or ten thoii.snnd times without 

 success, and at the end of all thoso trials ho would have no better 

 chance of succeeding in a fresh series of trials than at first com- 

 mencing. Wo cannot recognise any reason why ho remarkable a 

 set of throws as twelve successive "tails" should ercr reward his 

 patience. 



Yet it is not difficult to show that, given only a sufficiently largo 

 number of trials, the really wonderful thing would be that twelve 

 snccessivo " tails " should /uil to be thrown. 



To simplify matters, let us conceive that instead of one person 

 making a series of trial-tossings, we have a largo number of 

 ]jcr80MS, each of whom is to toss until head ap]'>ear8. Let us set 

 the number at one million. It is obvious that when each of these 

 million persons has tossed his coin once, about one-half will 

 have thrown tail. Say half exactly, for con»-enience of compu- 

 tation ; since, at any rate, we cannot regard it as a very wonderful 

 circumstance if as many as 500,000 of the million toss tail. 

 These 500,000 are now to toss again. About one-half will 

 again toss " tail." Say as before, exactly one-half. The 250,000 

 who have tossed tail twice toss it yet again ; and about 125,000 toss 

 "tail" a third time. Then the 125,000 toss a fourth time, and 

 aliout 02,500 toss tail a fourth time. So about 31,250 toss " tail " 

 a fifth time running; about 15,635 a sixth time; about 7,H12 a 

 seventh time ; about 3,00C an eighth time ; about 1,953 a ninth 

 time ; about 976 a tenth time ; about 188 an eleventh time ; about 

 241 a twelfth time ; about 123 a thirteenth time ; about C2 a four- 

 teenth time ; about 31 a fifteenth time; about IG a sixteenth time; 

 say 8 a seventeenth time ; i an eighteenth time ; 2 a nineteenth 

 time ; and one a twentieth time. When we get among these 

 smaller numbers we feel less confident of the result ; but among 

 the larger numbers, though we can by no means feel certain as 

 to the exact number of " heads" and " tails" that would be tossed, 

 ■we feel tho utmost confidence as to the general character of 

 the result. Thus, supposing 31,000 had tossed "tail" five times 

 running; then it would be a highly improbable thing that less than 

 IJ.OOO or 15,000 out of the 31,000 would toss "tail" on the next 

 trial. And even as respects the smaller numbers there would be at 

 least as fair a chance of as many " tails " being tossed as the above 

 reckoning assigns, as the contrary. So that, though a first, or 

 second, or third trial with our million tossers failed to give one 

 person, at least, who tossed " tail " twenty times in succession ; yet 

 a few successive trials (each trial including all the million persons) 

 would undoubtedly insure this seemingly incredible result, that 

 twenty successive tojsings of a coin could give an identical result.* 

 As for merely twelve successive " tails," we might be sure of 

 getting upwards of a hundred instances of that sort on the very 

 first trial. 



If we calculate how much would be paid on the lottery after 

 one of these sets of a million tossings, wo shall at once begin to see 

 why each iosser should pay a large sum for his chance. Instead of 

 doing this directly, let us begin with the case of a few tossings, and 

 estimate the effect of increasing the number of trials — assuming, 

 for convenience, that exactly half those who toss in any case, toss 

 "head," the other half tossing "tail." This assumption does not 

 influence the reasoning, because it is clear that if more than half 

 toss cither head or tail, it is as likely that more tails than heads as 

 that more heads than tails will be tossed. 



If there are four persons, two toss " head" and receive £2 each, 

 or ,£1 in all. On the second trial, one tosses " head " and receives 

 £1. On the third, say the one tosser left throws "head," and 

 receives £8. The money to be divided between the four persons is 

 thus, £16; or an average of £i to each. 



If there are eight persons, four toss "head" at the first trial, 

 and receive £2 each, or £8 in all ; two toss " head " at the second 

 trial, and receive ,£ I each, or £8 in all ; one tosses " head " at the 

 third trial, asd receives £8; the last tosses "head" at the last 

 trial (say), and receives £16. In all, the sum of £10 is to be paid 

 to these eight persons, or an average of £5 to each. 



In like manner, if there are sixteen persons, eight will get among 

 them £16 ; four will get among them another £16 ; two will receive 

 a third £16 ; one will get £16 ; and the last £32 ; or £96 in all will 

 have to be divided among sixteen persons, that is, an average of £6 

 to each. 



• In ten successive trials with our million of tossers, the odds arc 

 more than 10,000 to 1 that 20 successive " tails " will he tossed. 

 And only 603,117 out of the million need take part in one trial to 

 give an even chance of tossing twenty successive heads. Do Mor- 

 gan's book says 70,000 ; tut there Biust be a misprint. 



