Q. 3.31 - A First Course in Probability

Question

Suppose that a nonmathematical, but philosophically minded, friend of yours claims that Laplace's rule of succession must be incorrect because it can lead to ridiculous conclusions. "For instance," says he, "the rule states that if a boy is $10$ years old, having lived $10$ years, the boy has probability$\frac{11}{12}$ of living another year. On the other hand, if the boy has an $80$-year-old grandfather, then, by Laplace's rule, the grandfather has probability $\frac{81}{82}$of surviving another year. However, this is ridiculous. Clearly, the boy is more likely to survive an additional year than the grandfather is." How would you answer your friend?

Step-by-Step Solution

Answer

Sometimes given a person's attributes, the likelihood that they will survive another year is not reliant on the number of years they have lived.

Unlike with coins, where the chance of a heads flip is independent of the number of prior flips for a given coin$C_i.$

Step: 1 Laplace rule:

The Laplace distribution is a continuous probability distribution named after the French mathematician Pierre-Simon Laplace. It is one of the first known probability distributions. This distribution, like the normal distribution, is unimodal (has just one peak) and symmetrical.

Step: 2 Laplace's rule equation:

A coin $C_i~$Specific person

Coing getting heads $P\left(H\mid C_i\right) ~$Probability living another year.

$n$ gets flipped  $~ n$ lived years.

By deducing Laplace's rule process,

$P\left(H\mid C_i{F}_n\right)=P\left(H\mid C_i\right).$

Step: 3 Conclusion:

That was not the case when it comes to humans and their life span. Will the person survive another year is determined by a variety of factors, including $F_n$, and is not determined by the individual's attributes ($C_i$).