324 
C IT A 
C H A 
C H A 
imtion of a herald, make challenge to this 
effect, viz. “ that if any man shall deny the 
king’s title to the crown, he is there ready to 
defend it in single combat, &c.” which being 
done, the king drinks to him ; and sends him 
a gilt cup, with a cover, full of wine, which 
the champion' drinks, and has the cup for his 
fee. 
CHANCE, is more particularly used for the 
probability of an event; and is greater or less, 
according to the number of chances by which 
it may happen, compared with the number of 
chances lay which it may either happen or fail. 
Thus, if an event has three chances to happen, 
and two to fail, the probability of its happening 
may be estimated A, and the probability of its 
failing A Therefore, if the probabilities of hap- 
pening and failing are added together, the sum 
will always be equal to unity. 
If the probabilities of happening and failing- 
are unequal, there is what is commonly called 
odds for, or against, the happening or failing, 
which odds are proportional to the number of 
chances for happening or failing. 
The expectation of obtaining any thing, is 
estimated by the value of that thing/multiplied 
by the probability of obtaining it. The risk of 
losing any thing, is estimated by the value of 
that thing, multiplied by the probability of 
losing it. If, from the expectations which the 
gamesters have upon the whole sum deposited, 
the particular sums -they deposit (that is, their 
own stakes) are subtracted, there will remain the 
gain, if the difference is positive; or the loss, if 
the difference is negative. Again, if from the 
respective expectations which either gamester 
has upon the sum deposited by his adversary, 
the risk of losing what he himself deposits is 
subtracted, there will likewise remain his gain 
or loss. 
If there is a certain number of chances by 
which the possession of a sum can be secured, 
and also a certain number of chances "by which 
it may be lost, that sum may be insured for that 
part of it, which shall be to the whole, as the 
number of chances there are to lose it, is to the 
number of all the chances. 
If two events have no dependence on each 
ether, so that p be the number of chances by 
which the first may happen, and q the number 
of chances by which it may fail ; and likewise, 
that r be the number of chances by which the 
second may happen, and s the number of chances 
by which it may fail : multiply p -f- q by r -f- >, 
and the product pr 4- qr ps -j- qt will contain 
ail the chances by which the happening or fail- 
ing of the events may be varied amongst one 
another. 
From what has been said, it follows, that if a 
fraction expresses the probability of an event, 
and another fraction the probability of another 
event, and these two events are independant, 
the probability that these two events will hap- 
pen, will be the product of the two fractions. 
Ex. 1. Suppose a person playing with a single 
die offers to wager, that he will throw an ace 
each time for two successive throws ; what pro- 
bability has he of succeeding? 
Solution. Suppose the wager £-36, and that, 
on throwing the first time, an ace did come up ; 
then, because there are six faces on the die, only 
one of which is right, his expectation on the 
second throw will be i_th of £.36, or 4- — £.6; 
but the probability of this event being also _Lth, 
the expectation, before tee first throw must ne- 
cessarily be 4-th of £. 6, or £. 1 . Therefore the 
probability of his losing the wager wiil be 
r 7~ i s s , 
I — x — — 1 — — =: from which 
it appears that no person ought to hazard such 
a wager, unless the value of 35 to I shall be laid 
against him. 
Ex. 2. A person offers to lay a wager of £. 1, 
that out of a purse containing ot -n counters, 
of which r? are black, and n white, he will, blind- 
folded, at the first trial, draw a white counter; 
and also that, out of another purse containing 
m and n counters, of which m are black, and 
n white, he will also, blindfolded, at the first 
trial, draw a white counter ; and that, if he fail 
in either trial, his wager shall he lost : — What 
probability is there that lie shall succeed ? 
Solution. If, as in the last example, he had 
already succeeded in the first trial, it would fol- 
low that his expectation on the second will be 
N 
— : but if the success of the first trial be a 
M N 
condition of obtaining this expectation, then the 
probability of so doing will be — — ; which 
1 3 b m -{—/i 
multiplied into that expectation, wiil give 
X . — , or : 
M -j- N 
-J- n 
, for the proba- 
-j-7Z x M X N 
bility required. 
Hence the probability of the happening of 
two inciependant events, will be equal to the pro- 
duct of the probabilities of their happening se- 
parately. Of course, if the two events are of 
the same kind, then the probability will be 
More generally : 
m -££] 2 m 2 -{• 2mn n 2 ’ 
I. Let the number of chances for the happen- 
ing of an event be a, and the number of chances 
for its failing be b ; then the probability of its 
happening once in any number of trials will be 
expressed by the series 
a -J- b 
ah 2 aP aP 
| — - j" » - — - — 
Cl -j- P] Cl -j- /j] 
aP 1 
&C. 
a 4- P 3 a 4" a 4" 7) a “f* b 
continued to as many terni3 as are equal to the 
number of trials given. Thus, if a — 1 , /> — 5, 
and the number of trials be 1, 2, 3, 4, 5, &c, then 
the probability will be -i- for one trial ; — + 
-5 11. . , I , 5 . 25 
— tor two trials; 1 — — 
Qfl J n 1 cia I 07 r. 
36 
91 
216 
36 ' 6 ‘36 
for three trials, and so on. 
216 
II. Things remaining as before, the probabi- 
lity of the event’s happening twice in any given 
number of trials, will be expressed by the fol- 
lowing series, continued to as many terms, 
’wanting one , as the number of trials given : — 
4 , 245 , 3 44 , 4 44 , 5a 2 P 
4- -f r— — - + ----- 4 - ~&c. 
a-\-b ' 2 ci- \r-b J ’ 
=A+ 10 - + 75 + S0 ° -t : ; t“. The,. 
oo i oi o ~ 1000“ — — - 1 J- nese 
36 
216 
1296 1 7776 ' 46656 
56 
being summed up, we have for two trials 
16 - „ 171 
1 
for three trials 
, for four , 
216 1296 
36 
for five 
1526 
7776 ’ 
„ . 12281 
for six , Sec. 
46636 
III. The probability of the event’s happening 
three times in any number of trials, will be ex- 
pressed by the series - a - — ~' K ‘ ^ - -i- 
a -j- b 1 5 a 4- P\ 4 
644 10 44 . 
— - . .. - — [— which is to be continued to 
a 4 - 5 > <7+7)' 
as many terms, wanting two, as is the number 
of terms given : thus a and 5, as before, ~ -f- 
15 
H- 
-l- J.T— which being summed tit» 
777 6 ' 46656 6 1 
1296 
for three trials = — — , for four = — , for five 
m 1 0 * c sQfc 
1296 
276 
7776’ 
&c. 
Generally a — chances for happening, 
b — ditto for failing, 
/ = number of times for producing 
the event in a given No. of trials, 
rt — number of trials, 
a 4" b — •> ; then the following series is 
’ universal : 
lb 
x 1 4- H 
/./+ 1.4 
1.2.4 
l.l _j_ 1./ 4. 2 .P 
L2.3.4 
l.l-\- I./ + 2./+ 3.4 
pa 
which se- 
1 . 2 . 3.4.4 
For when l ~ 1, the series will be 
a ab . , - 
— 7 - — — . &c. as in the nrst case : 
a + b^ jqrpd 
1—2, the series will be 
a 2 245 . 
rr - 4 ' - T . 777 — , &c. as in the second ease 2 
747 2 7 + 7 3 
/ = 3, the series will be 
4 3 45 .... 
rwr 7 r~ + occ. as in the third case 5 
447 3 4+7 + 
and so on universally. 
Remark. The above series must be continued., 
to so many terms exclusive of the common mul- 
J 
tiplicator — , as are denoted by the number 
n — / -f 1. 
And, for the same reason, the probability of 
the contrary, that is, of the event’s not happen- 
ing so often as / times, making « — / + 1 — 
l)P 
will be expressed by the series — X 1 + 
_L /•/* ~} ~ 1 - aa 1 P'P + I- / + 2.rt 3 
1 1.2 .ss ' 1.2. 3. 4 
ries is to be continued to so many terms, exclu- 
sive of the common muitiplicator, as are denoted 
by the number /. 
Either series may be used, according as / is 
, n 4- J 
less or greater than — ' — . It is to be observed 
of these series, that they are both derived from 
the same principle : for, supposing two adver- ■ 
sarieSjAand B, contending about the happening- 
of that event, which has every time a chances 
to happen, and b chances to fail; that the chances 
a are favourable to A, and the chances b to B, 
and thatfA should lay a wager with B that his 
chances should come up / times in n trials; then, 
by reason B lays a wager to the contrary, he 
himself undertakes that his own chances shall, 
in the same number of trials, come up n — / -J- I 
times ; and therefore, if in the first scries, we 
change l into n — / -j- ], and vice versa; and 
also write b for a, and a for b; the second series 
will be formed. See Nature and Laws of Chance, 
by Thomas Simpson ; Demoivre on Chances; 
Dodson’s Mathematical Repository, vol. ii. p. 82, 
See. 
Chance-medley, in law, is (he acci- 
dental killing of a man, not wholly without 
the killer’s fault ; it is also called manslaugh- 
ter by misadventure, for which the offender 
shall have his pardon of course. 6 Ed. •!. c. 9. 
Jlut here is to be considered, whether the 
person who commits this manslaughter by 
chance-medley was doing a lawful thing; 
tor if the act -was unlawful it is felony. 
Chance-medley is properly applied to such 
killing as happens for self-defence in a sudd on 
rencounter. 4 Black. 183. 
