110 Proceedings of the Royal Society 
Problems discussed by Boole in the “ Laws of Thought 
1 . The probability that one or both of two events happen is p, 
that one or both of them fail is q. What is the probability that 
only one of these happens ? 
xy + x(\ - y) + (1 - x)y = p , 
x{l ~ V) + (1 ~ x) y + (1 I x) (1 - y) =■ q , 
it is required to find 
X{± - v) + y( 1 - X). 
Let 
<1 - y) + y(l-x) = a + b{xy + x(l -y) + ( 1 - x)y] 
+ c{x( 1 -y) + (f -x)y+( 1 -x)(\-y)} . 
Let x = 1 y = 1 then 0 = a + b 
„ x = 1 y = 0 „ 1 = a + + c 
„ x — 0 y = 1 ,, 1 = a + b + c 
„ x = 0 y = 0 „ 0 = a + c . 
We have four equations, but two of them are identical. When 
solved — 
a = - 1 , & = 1 , c = 1 , 
a(l - y) + y(l - x) = - 1 + 
This method by indeterminate coefficients serves to indicate 
whether a problem is determinate. For example, investigate the 
first problem by its means — 
y — a + bx + cxy . 
Then 
1 =rt + & + C, 
0 = a + b , 
1 = a 
0 
“ a }i 
= a J 
0 
0’ 
b = - 1 
c = 1 . 
0 0 J 
y = q ~ qX + xy 
0 
= xy + o i 1 - x )- 
2. The probabilities of two causes and A 2 are a and b respec- 
