The Social Security Administration has a table that shows the probability that a person of a certain age dies before his/her next birthday here:
[https://www.ssa.gov/oact/STATS/table4c6.html](https://www.ssa.gov/oact/STATS/table4c6.html)
Can I use the data there to compute the probability of death of a person of a certain age anytime in the next two years by a calculation with two probabilities in the table?
For example the table shows that a 118 year old male has about an 85% chance of dying before the year is out while a 119 year old has a 90% chance. Naively I had hoped to simply add probabilities but clearly this can’t be because then a 118 year old would have a 175% percent chance of dying in two years.
In: Mathematics
You don’t straight add percentages that isn’t how they work as they are basically division so can’t be added like that. So an 85% death rate means that if you take 100 people only 15 will survive, now if you take those 15 people and have a 90% death rate 1.5 of them will survive so the combined survival rate is 1.5% or a death rate of 98.5%.
It would be the probability of dying at 118 plus the probability of dying at 119, with the precondition that you didn’t die at 118 (so 100% – 85% = 15% chance of not dying)
So it would be P(118) + (15%)(P(119))
Or .85 + (.15 x .9) = .985 or 98.5%
But this also isn’t the probability that you’ll die in that specific age, it’s the probability that you’ll die at or before that age, since the probability of even making it to 118 is really really low.
You have to multiply the chance of living. A 118-year old has a 15% chance of living one more year, and a 0.15 * 0.1 = 0.015 chance of living two.
The way to do this is by multiplying conditional probabilities.
Another way of saying those stats, is that a 118-year-old man has a 15% chance of survival, and a 119-year-old man has a 10% chance of survival. To compute his odds of survival for 2 years, you multiply: .15 x .1 = 0.015 = a 1.5% chance of living through those 2 years.
You can’t add up percentages like that.
The simple is not to not calculate the chance of dying but the of surviving both years.
85% of death changes are a 15% life change. 90% of death is 10% live chance.
If you multiply them together so 0.15*0.10=0.015 = 1.5% You can write is as (1-0.85)*(1-0.9)=0.015 or directly as 1-(1-0.85)*(1-0.9)=0.985
So a person that is 118 has 1.5% chance of surviving 2 years so a 98.5% chance of dying.
You can do that with the death rate to but is a bit harder. You have to calculate the chase of dying each year by it selfe and add them together. Dying year 1 is simple but dying in year 2 requires you to survive the year.
* The chance of dying in the first year is 85%.
* Chance of dying the second year is 90% if you survive the first year so (chance of surviving year 1) * (chance of dying year two)= (1-0.85)* 0.9
So the result is 0.85+(1-0.85)* 0.9= 0.985 = 98.5 % chance of dying
This will be increasingly hard for each year you include bit the chase of living is just an extra term per year
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