Select exercises for young proficients in the mathematicks ... / By Thomas Simpson.

  • Simpson, Thomas, 1710-1761.
Date:
1752
Catalogue details

Licence: Public Domain Mark

Credit: Select exercises for young proficients in the mathematicks ... / By Thomas Simpson. Source: Wellcome Collection.

    333/348 (page 319)
    for as long as They both have a Chance of con¬ tinuing together. Hence the whole Probability* depending on the ift, 2d, 3d* 4th, &c. Year (by adding ~ to each of the Values firft found) appears to be 2 ab 3 2 ab9 lab JJ— &c. refpedtively : Therefore the Sum of all Thefe, lab continued to as many Terms (n) as there are Years pro¬ posed, will, evidently, be the true Value of the Proba¬ bility required. But the Sum of n Terms of the Series* I, 3, 5, 7, &c. is -=l nx: Therefore the required Probability is alfo truly exprefled by — But here we mull not omit to take Notice, that this Conclufion holds, only, when (n) the Number of Years is lefs than the Time that A and B have a Chance of living together : For, if n be greater than a, and the Age of A, at the fame time, exceeds That of B ; then, belides the Probability of Receiving the Sum 5, in the Time (a) during which A hath a Chance of living, there is a further Chance, or Probability, de¬ pending on the Contingency of receiving it, or of B's dying, after the Expiration of the faid Time : Which (as the Life A is then out of the Queftion) will confe- quently be exprefled by ^And this added to -—■ f b iab found above (by writing a for »), gives 'in-a ~~~tT for the true Meafure of the required Probability in this Cafe, where n is greater than a. Now, to apply thefe Concluflons to the Problem in Queftion* let c denote the Complement of the Life C : Then, a^ the Probability of his receiving the Rent of any propofed Year, is compounded of the Probability of his living till then, and of the Probability that both A and B are extinct, with the Reftri&ion of A’s dying firft (as above determined) ; it is manifeft that the prefent Value /