20   Life Contingencies

It should be noted that ,,PTy.

= npxy +npx (1— nPY) +nPy (1 — nPx) = npx +nPy — npx y.

3. Again, the probability that (x) will die in the nth year is

_ dx~-n—1 _   ,/,   ,/,   ,/,

n—1 lqx   1   — n—iPxQx--n—1 —n—iPx—nPx•

x

The probability that (x) will not die in the nth year is

1—n—1l qx=lx—d }n—1=(1—n—lpx)+npx= In—lQx+nPx, which is the probability that (x) dies either during the first (n -1) years or after n years.

The product

[n—1 qx+(1 n—1 I qx)] [n—1 Iqy+(1 n—1 I qY)]

is unity and can be expanded as

n—1 Igx•n—1 qY+n—1 Igx(1 n—1 I qy)+n—1Igy(1 n—1 Iqx)
+(1—n—1 qx)(1—n—1 I qy)•

The meaning of each of these four terms should be carefully noted.

1. Both (x) and (y) may die in the nth year from now,

2. (x) may die in the nth year and (y) in some other year.

3. (y) may die in the nth year and (x) in some other year, and

4. neither of them may die in the nth year.

Notice also that

n—1 I qx. n—1 I qy —n—lpxy • qx+n—1 : y+n—1•

4. If we have three lives, one aged x, one aged y and one aged z, note the following probabilities concerning the next n years.

1.  That all will survive n years: ,,px. npy. npx which is written npxyz•

2. That none will survive n years:

(1 —„px) (1 —net'y) (1 —npz).

3. That at least one death will occur: 1—npxyz.