ON PROBABILITY. 11 
We pis 
oo un Ex. 11. Let a jury be composed of n jurymen, and let p be the probability 
that each juryman separately will give aright decision, g the probability that he 
will give a wrong decision. The probability of a unanimous verdict, that is, 
that they all will voluntarily give a wrong or all a right decision, is pT 
° ) (p+9) 
if p= 5510 and n = 12, this is equal to £322 F 555%, or about 13, 
i This probability must not be confounded with the probability, after an 
unanimous decision has been given, that it is a correct one, which is a very 
{Qin the different question. 
Ex. 12. Let A and B be two gamesters, and let p be the probability of 
wi Pi A’s winning a game, and ¢ the probability of B’s winning a game ; required 
40 aud the probability of A’s winning m games out of m + n, the set being sup- 
Qi any posed to finish as soon as A has won m games, 
3 Q, Without The probability of A’s winning the first m games running, is p™. The 
Kk: Whish ae probability of A’s winning m games out of m 4 1,is (m + 1) p”g. The 
or probability of A’s winning the first ; games, and B the (m + 1)* out of 
: m + 1 games, is p™ ¢; therefore, the probability of A’s winning the set in 
exactly m + 1 games is (m +1) p"q— pq, or (m +1 = 1) p"gq 
= = m p™gq, and the probability of A’s winning the set in not more than 
\ m + 1 games is p™ + m p™ . q. 
0 find what The probability of A’s winning m games out of m —}- 2, in any order, is 
ary 10 fake (m-+2).(m+1) 
i: zl 
but the probability of A’s winning m ‘games in the first m1 is 
(m + 1) . p™q, and the probability of B’s winning the {m + 2)" game is 
g ; therefore, the probability of A’s winning m games in the first m + 1 out 
of m 4 2 games is (m1). p"¢*; and the probability of A’s winning 
the set exactly in m -|- 2 games is 
(m+2).(m+1) borg 2 FD pe 
! 2 f= 1) pt pm =e i Lng; 
and the probability of A’s winning the set in not more than m + 2 games is 
7 " m.(m+1D 
7 b+ 8 Pq) op mri 
The same reasoning may be applied to the general term : thus the pro- 
bability that A will win 7m games out of m + 7 in any order 
mtn). etnol),. 20) 
RP Bang Ih iFL ML) Zt 
The probability that A will win m games out of m --n — 1, and that B 
will win the (m + n)* 
(rr =D. rte ="). 2 
nf BN Trl. = et 
hy Therefore the probability that A will win m games in exactly m + n 
A cames 
ba = ZA Naga) SR RTE ; 
BSS Zeer a 1: 2a (1) n Pq 
cnt Dinan rb —1 , , 
zs 1 2 2 0 0 23 8 es 2° 94