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The Gaming Table-Vol.2-Part23_Andrew Steinmetz(1816-1877)_English Literature(文学名著)-第2页

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happen, and 2 to fail, then the fraction 3/5 will fairly

represent the probability of its happening, and may be taken to



be the measure of it.

The same may be said of the probability of failing, which will



likewise be measured by a fraction whose numerator is the number

of chances whereby it may fail, and the denominator the whole



number of chances both for its happening and failing; thus the

probability of the failing of that event which has 2 chances to



fail and 3 to happen will be measured by the fraction 2/5.

The fractions which represent the probabilities of happening and



failing, being added together, their sum will always be equal to

unity, since the sum of their numerators will be equal to their



common denominator. Now, it being a certainty that an event will

either happen or fail, it follows that certainty, which may be



conceived under the notion of an infinitely great degree of

probability, is fitly represented by unity.



These things will be easily apprehended if it be considered that

the word probability includes a double idea; first, of the number



of chances whereby an event may happen; secondly, of the number

of chances whereby it may either happen or fail. If I say that I



have three chances to win any sum of money, it is impossible from

the bare assertion to judge whether I am likely to obtain it; but



if I add that the number of chances either to obtain it or miss

it, is five in all, from this will ensue a comparison between the



chances that are for and against me, whereby a true judgment will

be formed of my probability of success; whence it necessarily



follows that it is the comparativemagnitude of the number of

chances to happen, in respect of the whole number of chances



either to happen or to fail, which is the true measure of

probability.



To find the probability of throwing an ace in two throws with a

single die. The probability of throwing an ace the first time is



1/6; whereof 1/ is the first part of the probability required.

If the ace be missed the first time, still it may be thrown on



the second; but the probability of missing it the first time is

5/6, and the probability of throwing it the second time is 1/6;



therefore the probability of missing it the first time and

throwing it the second, is 5/6 X 1/6 = 5/36 and this is the



second part of the probability required, and therefore the

probability required is in all 1/6 + 5/36 = 11/36.



To this case is analogous a question commonly proposed about

throwing with two dice either six or seven in two throws, which



will be easily solved, provided it be known that seven has 6

chances to come up, and six 5 chances, and that the whole number



of chances in two dice is 36; for the number of chances for

throwing six or seven 11, it follows that the probability of



throwing either chance the first time is 11/36, but if both are

missed the first time, still either may be thrown the second



time; but the probability of missing both the first time is

25/36, and the probability of throwing either of them on the



second is 11/36; therefore the probability of missing both of

them the first time, and throwing either of them the second time,



is 25/36 X 11/36 = 275/1296, and therefore the probability

required is 11/36 + 275/1296 = 671/1296, and the probability of



the contrary is 625/1296.

Among the many mistakes that are committed about chances, one of



the most common and least suspected was that which related to

lotteries. Thus,supposing a lotterywherein the proportion of



the blanks to the prizes was as five to one, it was very natural

to conclude that, therefore, five tickets were requisite for the



chance of a prize; and yet it is demonstrable that four tickets

were more than sufficient for that purpose. In like manner,



supposing a lottery in which the proportion of the blanks to the

prize is as thirty-nine to one (as was the lottery of 1710), it



may be proved that in twenty-eight tickets a prize is as likely

to be taken as not, which, though it may contradict the common



notions, is nevertheless grounded upon infallible demonstrations.

When the Play of the Royal Oak was in use, some persons who lost



considerably by it, had their losses chiefly occasioned by an

argument of which they could not perceive the fallacy. The odds



against any particular point of the ball were one and thirty to

one, which entitled the adventurers, in case they were winners,



to have thirty-two stakes returned, including their own; instead

of which, as they had but twenty-eight, it was very plain that,






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