Probabilities of Death and Survival   21

4. That at least one life will survive:

1—(1 —npx) (1 —npy) (1—npz).

5.  That not more than one will die:

npxy+nP 2+npzx -2npxyz.

6. That not more than two will die: the same as (iv).

7. That not more than one will live:

1 — (npxy'+npyz +npzx) +2npxyz•

8. That not more than two will live: the same as (iii).

(ix) That the survivor will live out the period: npxyz = the same as (iv) or (vi).

(x) That the survivor will not live out the period: 1—npxyz =the same as (ii).

The probability that (x) will die during the nth year from now is ,i_1px•qx+n_1=n–1px(1—px-F,a_1)=n_1px—npx, therefore the probability that (x) will die before the nth year begins, that (y) will die in the nth year, and (z) survive the n years is

(1 —n–lpx) (n–lpy npy)npz•

5. It is a great advantage to use symbols for well-defined cases in the solving of most problems involving probabilities. Each symbol as used above is a function of all definitely placed prefixes or suffixes that may be attached to it, so that each symbol when completed represents a numerical value, depending on the positions and numerical values of the prefixes or suffixes that it sup-ports..

The numerical values of lx and dx remain the base from which numerical values of the symbols representing probabilities of life and death more complicated than simple qx can be produced. Hence it is important that the student should be able to turn given symbols into different forms especially those forms which involve the lx and dx columns.

The student must use much care in the production of his symbols. The ordinary operations of elementary mathematics and his common sense should be at his command in order to analyse the component parts of the problems he has to tackle or