on the happening of that event. Therefore if the expectation on

an event be divided by the value of the thing expected, on the



happening of that event, the quotient will be the probability of

happening.



Example I. Suppose two horses, A and B, to start for L50, and

there are even bets on both sides; it is evident that the present



value or worth of each of their expectations will be L25, and the

probabilities 25/50 or 1/2. For, if they had agreed to divide



the prize between them, according as the bets should be at the

time of their starting, they would each of them be entitled to



L25; but if A had been thought so much superior to B that the

bets had been 3 to 2 in his favour, then the real value of A's



expectation would have been L30, and that of B's only L20, and

their several probabilities 30/50 and 20/50.



Example II. Let us suppose three horses to start for a

sweepstake, namely, A, B, and C, and that the odds are 8 to 6 A



against B, and 6 to 4 B against C--what are the odds--A against

C, and the field against A? Answer:--2 to 1 A against C, and 10



to 8, or 5 to 4 the field against A. For

A's expectation is 8



B's expectation is 6

C's expectation is 4



----

18



But if the bets had been 7 to 4 A against B; and even money B

against C, then the odds would have been 8 to 7 the field against



A, as shown in the following scheme:--

7 A



4 B

4 C



----

15



But as this is the basis upon which all the rest depends, another

example or two may be required to make it as plain as possible.



Example III. Suppose the same three as before, and the common

bets 7 to 4 A against B; 21 to 20 (or 'gold to silver') B against



C; we must state it thus:--7 guineas to 4 A against B; and 4

guineas to L4, B against C; which being reduced into shillings,



the scheme will stand as follows:--

147 A's expectation.



81 B's expectation.

80 C's expectation.



----

311



By which it will be 164 to 147 the field against A, (something

more than 39 to 35). Now, if we compare this with the last



example, we may conclude it to be right; for if it had been 40 to

35, then it would have been 8 to 7, exactly as in the last



example. But, as some persons may be at a loss to know why the

numbers 39 and 35 are selected, it is requisite to show the same



by means of the Sliding Rule. Set 164 upon the line A to 147

upon the slider B, and then look along till you see two whole



numbers which stand exactly one against the other (or as near as

you can come), which, in this case, you find to be 39 on A,



standing against 35 on the slider B (very nearly). But as

164/311 and 147/311 are in the lowest terms, there are no less



numbers, in the same proportion, as 164 to 147,--39 and 35 being

the nearest, but not quite exact.



Example IV. There are four horses to start for a sweepstake,

namely, A, B, C, D, and they are supposed to be as equally



matched as possible. Now, Mr Sly has laid 10 guineas A against

C, and also 10 guineas A against D. Likewise Mr Rider has laid



10 guineas A against C, and also 10 guineas B against D. After

which Mr Dice laid Mr Sly 10 guineas to 4 that he will not win



both his bets. Secondly, he laid Mr Rider 10 guineas to 4 that

he will not win both his bets.



Now, we wish to know what Mr Dice's advantage or advantage" target="_blank" title="n.不利(条件)；损失">disadvantage is,

in laying these two last-mentioned wagers.



First, the probability of Mr Sly's winning both his bets is 1/3

of 14 guineas; and Mr Dice's expectation is 2/3 of 14 guineas, or



L9 16s., which being deducted from his own stake (10 guineas),

there remains 14s., which is his advantage" target="_blank" title="n.不利(条件)；损失">disadvantage in that bet.



Secondly, Mr Rider's expectation of winning his two bets is 1/4,

and, therefore, Mr Dice's expectation of the 14 guineas, is 3/4,



or L11 0s. 6d., from which deduct 10 guineas (his own stake), and

there remains 10s. 6d., his advantage in this bet,--which being



deducted from 14s. (his advantage" target="_blank" title="n.不利(条件)；损失">disadvantage in the other), there remains

3s 6d., his advantage" target="_blank" title="n.不利(条件)；损失">disadvantage in paying both these bets.



These examples may suffice to show the working of the system;

• account [ə´kaunt] vi.说明 vt.认为 n.帐目 (初中英语单词)


• complain [kəm´plein] vi.抱怨，叫屈；控诉 (初中英语单词)


• undertake [,ʌndə´teik] vt.从事；承担；担保 (初中英语单词)


• actually [´æktʃuəli] ad.事实上；实际上 (初中英语单词)


• advantage [əd´vɑ:ntidʒ] n.优势；利益 (初中英语单词)


• doctrine [´dɔktrin] n.教义；主义；学说 (初中英语单词)


• whenever [wen´evə] conj.&ad.无论何时 (初中英语单词)


• maintain [mein´tein] vt.维持；保持；继续 (初中英语单词)


• fortunate [´fɔ:tʃənət] a.幸运的，侥幸的 (初中英语单词)


• running [´rʌniŋ] a.奔跑的；流动的 (初中英语单词)


• possibility [,pɔsə´biliti] n.可能(性)；希望；前途 (初中英语单词)


• therefore [´ðeəfɔ:] ad.&conj.因此；所以 (初中英语单词)


• succession [sək´seʃən] n.继任；继承(权) (初中英语单词)


• amount [ə´maunt] n.总数；数量 v.合计 (初中英语单词)


• proportion [prə´pɔ:ʃən] n.比率 vt.使成比例 (初中英语单词)


• application [,æpli´keiʃən] n.申请；申请书；应用 (初中英语单词)


• player [´pleiə] n.游戏的人；选手 (初中英语单词)


• constantly [´kɔnstəntli] ad.经常地；不断地 (初中英语单词)


• voyage [´vɔi-idʒ] n.&vi.航海；航程；旅行 (初中英语单词)


• knowing [´nəuiŋ] a.会意的，心照不宣的 (初中英语单词)


• beaten [´bi:tn] beat 的过去分词 (初中英语单词)


• evident [´evidənt] a.明显的，明白的 (初中英语单词)


• scheme [ski:m] n.计划；阴谋，诡计 (初中英语单词)


• supposed [sə´pəuzd] a.想象的；假定的 (初中英语单词)


• likewise [´laikwaiz] ad.同样地；也，又 (初中英语单词)


• working [´wə:kiŋ] a.工人的；劳动的 (初中英语单词)


• system [´sistəm] n.系统，体系，制度 (初中英语单词)


• disadvantage [,disəd´vɑ:ntidʒ] n.不利(条件)；损失 (高中英语单词)


• saying [´seiŋ, ´sei-iŋ] n.言语；言论；格言 (高中英语单词)


• signify [´signifai] vt.表示 vi.有重要性 (高中英语单词)


• whilst [wailst] conj.当…时候；虽然 (高中英语单词)


• perfectly [´pə:fiktli] ad.理想地；完美地 (高中英语单词)


• concerned [kən´sə:nd] a.有关的；担心的 (高中英语单词)


• equivalent [i´kwivələnt] a.相等的 n.同等物 (高中英语单词)


• incredible [in´kredəbəl] a.不能相信的；惊人的 (高中英语单词)


• continually [kən´tinjuəli] ad.不断地，频繁地 (高中英语单词)


• dealing [´di:liŋ] n.交易；来往 (高中英语单词)


• reasoning [´ri:zəniŋ] n.推理，评理 a.推理的 (高中英语单词)


• expectation [,ekspek´teiʃən] n.期待(望)；预期 (高中英语单词)


• probability [,prɔbə´biliti] n.或有；可能性 (高中英语单词)


• namely [´neimli] ad.即，也就是 (高中英语单词)


• suffice [sə´fais] vt.使满足 vi.足够 (高中英语单词)


• prodigious [prə´didʒəs] a.惊人的；巨大的 (英语四级单词)


• winning [´winiŋ] n.&a.胜利(的) (英语四级单词)


• adversary [´ædvəsəri] n.敌手，对手 (英语四级单词)


• considering [kən´sidəriŋ] prep.就…而论 (英语四级单词)


• varied [´veərid] a.各种各样的 (英语四级单词)


• superstitious [,sju:pə´stiʃəs] a.迷信的 (英语四级单词)


• whatsoever [,wɔtsəu´evə] (强势语)=whatever (英语四级单词)


• happening [´hæpəniŋ] n.事件，偶然发生的事 (英语四级单词)


• requisite [´rekwizit] a.需要的；必要的 n.必需品 (英语四级单词)


• inherent [in´hiərənt] a.固有的，天生的 (英语六级单词)


• intervention [,intə´venʃən] n.干涉；调停；插入 (英语六级单词)


• lottery [´lɔtəri] n.抽彩，抓阄，彩票 (英语六级单词)


• guinea [´gini] n.几尼(英国旧金币) (英语六级单词)


• prevailing [pri´veiliŋ] a.占优势的；主要的 (英语六级单词)


• holding [´həuldiŋ] n.保持，固定，存储 (英语六级单词)


• respecting [ri´spektiŋ] prep.由于；鉴于 (英语六级单词)


• applied [ə´plaid] a.实用的，应用的 (英语六级单词)


• secondly [´sekəndli] a.第二(点)；其次 (英语六级单词)


• deduct [di´dʌkt] vt.扣除；演绎(推理) (英语六级单词)