‘ CN PROBABILITY. 
The probability of the obverse once and the reverse once in any order is 
1, because the second and third cases favour the production of this event; 
and the probability of the obverse arising at least once is #, because the 
first, second, and third cases are favourable to this event. 
Ex. 2. Again, suppose of two bags one contains 5 white balls and 2 black, 
and the other 7 white balls and 3 black. The number of cases possible in 
one drawing from each bag is (542) X (7-3) or 7 x 10, because every 
ball in one bag may combine with every ball in the other, which cases, if we 
are ignorant of any cause favouring the appearance of a white rather than a 
black ball, are all similarly circumstanced. 
The number of cases which favour the drawing a white ball from both is 
5 x 17, for every one of the 5 white balls in one bag may combine with every 
one of the 7 white balls in the other. For a similar reason, the number of 
cases which favour the drawing a white ball from the first bag and a black 
ball from the second, is 5 X 3; a black ball from the first bag and a white 
ball from the second, is 7 xX 2; and a black ball from both is 3 x2. There- 
fore, 
Puy bs the probability of drawin hite ball from both 
rt mre £2 ee } abili ing a whi r . 
GCIFHEFH ar y Hild 
N DXB ms {a whiteball from the firstand 
C+D 1a | ablack ball from the second. 
id 7x2 ad (a black ball from the first and 
G+-Ha +3 5° 0° co | awhite ball from the second. 
3x2 3 
Erm ° e . » . 1 kb 11 from both. 
GLH G+ 35 a black ball fr 
The probability of drawing one white ball, without reference to the bag 
from which it comes, is 
h3+2x7 29 
G+2)x T+3) 10 
for both the second and third cases favour the production of this event. 
The probability of drawing at least one white ball is 
BX74+5xXx3+2x%x7 32 
G+ x13. . 35 
for the first, second, and third cases favour the production of this event. 
Let the number of white and black balls in each bag be the same, say 5 
white and 2 black, then the probability of drawing 
: 5x5 25 
a white ball.fiomboth . ,.T 0 a nT “B12 x (512) 49 
Tn Ba 
a white ball from the firstand a black ball from the secon = 512) (012) 19 
a black ball from the first and a white ball from th d ax 1 
standa ¢ = 
white ball from the secon G12) G12) 19 
2 
a black ball from both . »..= HEE > |, 00 sa Bl = 4 . 
G+2) (B+2) 49 
Ex. 3. Two dice are thrown; required the probability that the sum of the 
numbers on the sides which fall uppermost, or the throw, is any given num- 
ber, say 7.