Examples   83

2. That only one of them will be dead n years hence.

3. That they will die in alphabetical order.

4. That A will die after B but during the life of C.

9. There are three lives, (x), (y), (z). Write down expressions for the following probabilities:

1. That they will not all be dead n years hence.

2. That they will not all be living n years hence.

3. That (x) and at least one other will be living it years hence.

4. That they will all live ten years and that (x) will die after (y) but before (z).

5. That (x) will die in the first year and be the first to die.

10. The probability that A will die within ten years is .2. The probability that A, B and C will all be alive ten years hence is .42. The probability that at least one of the three will be alive ten years hence is .985. Find the probability that A and B alone will be living at the end of the tenth year.   .

11. Given I„Qx.x+n =.9 and „p, =.5, find the probability that of three lives each aged x

1. at least two will die between ages x+n and x+2n

2. the third death will take place between the ages x+n and x+2n.

12. Express   Qxy„   in terms of probabilities de-
termining at the first death.

13. Differentiate ,,px (1) with regard to x (2) with regard to n.

ex.ex+l.ex+2    ex+n_i

= npx

14. Prove

(1-hex+1)(1+ex+2) .. (1+ex+n)