Deferred Survival Probabilities from Mortality Rates
Summary
The document explains how to calculate the probability that a life survives across multiple years using annual mortality-table probabilities. If q at age x is the probability of dying before the next birthday, then p at that age is its complement, the probability of surviving to the next birthday. Survival to a later age is found by multiplying the relevant one-year survival probabilities.
The apparent dependence between survival events does not invalidate multiplication. The probability of surviving to age 52 can be written as the probability of surviving to age 51 multiplied by the conditional probability of surviving from 51 to 52, given survival to 51. The table’s p values supply these age-specific conditional probabilities. The example illustrates the basic probability rule, but it does not discuss uncertainty in estimated mortality rates or variations in mortality across populations.
Key ideas
- An annual survival probability is the complement of the corresponding mortality probability.
- The chance of surviving several years is the product of the annual conditional survival probabilities.
- The multiplication follows from the conditional probability rule and does not assume independent survival events.
- Mortality tables provide probabilities specific to each attained age.
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# Deferred mortality probabilities (mortality table)
# Deferred mortality probabilities (mortality table)
My question has to do with drawing correct conclusions regarding deferred mortality probability from a mortality table.
I am looking at the table below (source). In it, the $q_x$ (2nd columns) is given, whilst the $p_x$ I've computed simply as $1-q_x$.
Goal
I would like to find an answer to the following question: for a life age 50 ($x$), what is the probability that this life will survive until 51 ($x+1$), 52 ($x+1$), 53 ($x+1$), 54 ($x+1$), 55 ($x+1$) ... That's my goal.
I've watched this (very good) youtube video that seems to offer a recipe for precisely this kind of calcs. However... I struggle with something conceptually quite fundamental in it.
Problem With Understanding
Let me use an example:
- This life surviving one year (i.e., reaching its 51st birthday) is easy, and the $p_{50}$ can be obtained directly from the table as 0.994692.
- However, how about this life surviving until its 52nd birthday? If I would follow the argument from the video (which, granted, is for someone surviving some time, and then dying, but I assume the argument/mechanism goes the same way for the question that I ask), I should say: this life surviving until 52 is equivalent to it surviving until 51, and then from 51 until 52, and this can be expressed as $p_{50} \cdot p_{51} = 0.994692 \cdot 0.994314$. This is, at leats, how I understand the argument from the video. But ... wouldn't this imply that the two events (surviving until 51 and surviving until 52) are independent? Clearly, in order to have a shot at surviving until 52 one must survive until 51, so to me the dependency is rather obvious ... Yet, the multiplication of the two probabilities - the $p_{50} \cdot p_{51}$ - and assuming that this is the answer, looks as if the two events were not dependent???
Ask
How should I approach the task at hand? What formula to use, and if the one with the "simple" product, how can this be justified - especially from the conditional probability standpoint, please?
## Answer by Robert (score 1)
https://quant.stackexchange.com/a/80108
Your calculation is fine. Let $X$ denote the age of death. You can calculate $P[X>52] = P[X>52 \text{ and } X > 51] = P[X>52 | X > 51]\cdot P[X>51] = p_{51} \cdot p_{50}$ using the basic property of conditional probability.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.