‘ 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.