Probabilities of Death and Survival   19

2. (x) may survive the n years and (y) fail to do so, the probability being,

nt'x I nQy=nh/~

t'x• Qy =npx(   npy) npx—npxy

=the probability that (x) will survive n years less the probability that the pair does so.

Or   InQY— I,,Qx• InQy= Qylnl—Qx   ~~/1

`Y

=the probability that (y) dies within the period less the probability that both of them do so.

Or   InQy— InQxy =Q'y —Q.zY ail

=the probability that (y) will die within the period less the probability that the survivor of (x) and (y) will so die.

Or Qxy 1 = the probability that of the three statuses, the life of x, the life of y, and the period of n years certain, the life of y will fail first and the period of n years elapse second.

3. (x) may die within the period and (y) survive it, the probabilities being as in (2) interchanging x and y.

4. Both (x) and (y) may die within the period, the proba-
      bility being   /

nQx• nQy=(1—npx) (1—npy)=(1—npx) n//~~ t'y(—n/~

t'x)

=the probability that (x) will die within the n years less the probability that he will be the only one to do so.

Or QX' nl =the probability that the survivor will die within the period.

Or Qxy nl +Qxy which is the probability that (x) will die

   t   t

before the n years pass but not first of the three statuses plus the probability that (y) will die before the n years pass but not first of the three statuses.

Or Qx nl —Qxy its +Qy n~ —Qxynl

=Qx nI +QY,—QxY = nQx+I,Q3,— ,,Qxy

= 1 —nPx+1npy (1 —npxy) = 1 — (nPx+nPY —,tpxy)

=1—npxy = the complement of the probability that the survivor will live n years.