PROBABILISM 



PROBABILITIES 



431 



to windward, supporting a weight which counter- 

 balances the pressure of the wind on the sail. 

 The sail resembles the ordinary lug-sail, and is 

 formed of mat. Slight variations from this form 

 are found, but the principle of construction is the 

 game. 



Probabilism. See CASUISTRY. 



Probabilities, CHANCES, or- the THEORY OF 

 AVEUACKS. To assign a number which measures 

 the probability of a future event may at first seem 

 impossible ; and yet the whole business of many 

 large; companies instituted in every civilised country 

 for the ' insurance ' or ' assurance ' of lives, &c. is 

 mainly based upon the methods of assigning such a 

 miml>er. When it is certain that a future event 

 will take place, or will not take place, a fixed num- 

 ber is selected for each case to indicate that then 

 the probability amounts to certainty : ami these two 

 measures are the limits of our scale. Will the sun 

 rise to-morrow morning in the east? Probability = 

 1, certainty in favour. Will full moon be seen to- 

 morrow morning in the east? Probability = 0, 

 certainty against. Between these two limiting 

 numbers, and 1, lies the number (a proper frac- 

 tion ) which measures the probability of any unde- 

 cided event. The number, then, by which we 

 mark the chance, or expectation, or probability of 

 anything occurring in the future, must be a fraction 

 like j, j*j, 4|j, or "273, and can never Vie so large as 

 1, which was fixed as the higher limit, certainty : 

 and by the fractional number assigned to any event 

 we can readily compare its probability with those 

 of other future occurrences. 



To assign the proper fraction to any future 

 event will, in general, imply knowledge of a large 

 number of similar events. Thus, in January, what 

 is the probability that on next 12th April the sun 

 will rise bright and unclouded ? Relying on the 

 constancy of nature and the doctrine of averages, 

 we consult the calendars and weather-notices of the 

 last .">') years, say, and find that in 17 of these the 

 result was favourable and in 33 unfavourable. On 

 these data the probability required is 5, rather 

 over J. In other words, the odds are nearly 2 to 1 

 against the event. The fraction J measures or 

 shows the probability that the event will not 

 happen. More generally, if any future event may 

 occur in 12 ways and fail in 15 ways, then the prob- 



12 

 ability of its occurring is - rs = i ; and the 



probability of failure, 



15 



2 + 15 



= f . In such a case 



^ - 



12+15 



the 27 ways are supposed to have each the same 

 chance of occurrence : and, since the event must 

 either happen or fail, the sum of the two probabilities 

 =certainty Le. $+4 = 1. Thus, if J is the chance 

 of an event, 1 - J = chance that it will not occur. 

 In a certain town only 4 days of May taking the 

 average of many years are rainless : what will lie 

 our chance of finding next loth May rainless ? 

 Chance = / r : and 1 - 3*1 = chance of having rain. 

 The principle involved in such simple solutions is 

 the foundation of the mathematical treatment of 

 chance or probability. Of all the occurrences, all 

 equally possible, which relate to a future event, if 



a are favourable and x unfavourable, then = 



a + x 



where stands for prolwibility of the event occur- 

 ring. Sometimes it is easier to find the probability 

 of the event failing, and subtract that result from 

 1 as in the examples just given. 



Out of 100 sailors who mutinied there were 10 

 ringleaders. If 2 are selected by lot for capital 

 punishment, find the chance that both will be ring- 



100. 99 



leaders. The total number of pairs is 



1.2 



-, and 



the number of pairs among the ringleaders is 



Hence chance required = 



10 . 9 100 . 99 



10.9 

 1.8' 



1.2 ' 1.2 

 i.e. the odds are 109 to 1 against the event. A bag 

 contains 5 sovereigns and 4 shillings : if a child is 

 asked to draw three coins at random, what is the 

 probability that 2 will be sovereigns and 1 a shilling? 

 Here the total number of groups of 3 which can be 



987 

 formed out of all the 9 coins is ^ j^-=, or 84, which 



1 . 2t , o 

 forms our denominator. Of the sovereigns there 



are ^-^ = 10 pairs, each of which may be drawn 



with each of the 4 shillings, giving 40 groups of 3, 

 which forms our numerator. Hence chance required 

 i* ir = ir ! i- e - the odds are 11 to 10 against the 

 event. 



Sometimes actual trial seems to throw discredit 

 on the mathematical measure of a chance. Thus, if 

 a die be thrown, the chance of a 5 or any other 

 number turning up must be i by our definition : 

 whereas a person may cast a die, say 20 times in 

 succession, with the result : ace, 4 times ; 6 and 4, 

 each 3 times ; 2 and 3, each 5 times ; 5 not at all. 

 How then explain the mathematical estimate? 

 Simply that 20 is much too small a number to take 

 an average from, and the result ' chance = i for 

 each side of the die ' refers to the most general case 

 possible i.e. a very large number or even an infinite 

 number of throws. Register for 10,000 throws, then 

 for 100,000 or 1,000,000, and the results would more 

 and more approximate to the mathematical result, 

 and prove that each side has chance = J the die 

 being of course a perfect cube. 



An important extension of the theory is that the 

 probability of two independent events botli occur- 

 ring is measured by the product of their separate 

 probabilities. Thus, if A's chance of passing a 

 certain examination is f and B's \%, then (1) the 

 chance that both will pass is } x ]{ = -fa i.e. the 

 odds are 7 to 5 against ; ( 2 ) the chance that both 

 will fail ! (1 - f) (1 - j|) - T*T ! (8) the chance 

 that A passes and B fails is J (1 - |J) = 55 ; and 

 (4) the chance that A fails and B passes is ( 1 - J) 

 r = li- By comparing these four results we see 

 that the last event is the most probable of all, the 

 odds l>eing 25 to 24 in favour of it. Moreover, these 

 results exhaust the possible alternatives of double 

 event, therefore the four probabilities should to- 

 gether amount to certainty : and j'j + jjj + fa + 



By the same principle we solve many useful 

 and curious problems. A town-council of 20, 

 12 Liberals and 8 Conservatives, have to 

 choose a deputation of 5 by ballot : find the 

 proliability that it will contain 3 Liberals and 

 2 Conservatives. Total number of groups of 5 is 



20. 19. 18. 17. 16 



. = , or 19. 3. 17. 16, which forms 



J.. ^ . O . 4 d 



our denominator. Number of groups of 3 from the 

 , or 2 . 11 . 10, and number of 



Liberals is ~ 2 ,' 



1.2.3 



.8.7 



ability is ' ' -.' , or 



iy . o . 17 . 10 



pairs of the Conservatives is - . or 4 . 7 ; therefore, 



multiplyings 2 . 1 1 . 10 . 4 . 7 = total number of groups 

 of 5 which fulfil the conditions; and required prob- 



In other words, 



the odds are 584 to 385, or more than 3 to 2 against 

 the event. 



When a person buys lottery tickets his chance 

 of success is found as in our opening paragraphs, 

 and if multiplied by the value of the money attain- 

 able the product is called his ' expectation. ' In this 

 connection may be noted an important distinction 



