107 
of Edinburgh, Session 1878-79. 
Let U = a succession of states of the atmosphere at a given place, 
an individual state being of the length of a day. 
x — containing a thunderstorm, 
y — containing a hailstorm. 
Then the data are — 
x = xy = q. 
Hence, by means of the theorem proved 
_ - 0 - 
y = 1 + o< 1 ~P)> 
• •• y>i, 
and < q + T ~ p • 
(2.) A says that B says that a certain event took place ; required 
the probability that the event did take place, p 1 and p 2 being A’s 
and B’s respective probabilities of speaking the truth. 
The solution of this problem recently gave rise to a great amount 
of discussion in the “Educational Times.” Ho fewer than four 
different solutions are given, viz. : — 
Todhunter — 
PiPi + (! - Pi) (! ~Pi)- 
Artemas Martin — 
PAPiPi + 0- -Pi) (! -Pi)}- 
Woolhouse and American mathematicians — 
lhP2- 
Cayley— 
P 1 P 2 + £(1 - Pi) (! - Pi) + *(1 - P) 0 - Pi)> 
where (3 is the chance, on the supposition of the incorrectness of A's 
statement, that B told A that the event did not happen, and 1-/3 
that he did not tell him at all. k is the antecedent probability. 
Let XJ = statements of A about B’s statements about an event 
taking place. 
x = which truly reported a statement by B , 
y = which truly reported the event. 
