Given age and gender of 2 people, what is the formula to figure out probabilities that a specific person outlives the other ?
I know this used to be on 3L way back when and is still probably covered in some lower tier exam. Any help is appreciated.
I doubt 3L really covered it. You could get close by summing t_p_x times (t_p_y)q_(y+t) from 0 to the end of the table.
That’s summing both surviving t years and then y dying in the next year. Slight overstatement since it doesn’t cover x and y both dying in that next year, and x could die first. (Slight improvements are possible to approximate that effect, but probably beyond the scope of 3L).
If this is for a real world calculation, that approach also assumes that the lifetimes of x and y are independent, often not valid (though close enough for many purposes)
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This is a real world example, and the lives are independent other than same environment. I am only looking for a reasonable estimate of the likelihood that a 50(M) outlives a 47(F).
My best guess is that 47(F) outlives 50(M) with 0.8 probability, but that is simply a WAG.
At any given age forward per SSA table, the current 50(M) seems to have about double the force of mortality as the current 47(F) up until ages 80(M) and 77(F), where that trend becomes a little less severe (at 90(M) and 87(F) it is closer to 1.5 as an example)
Without looking at the SSA table or trying to do my calculation, it seems to me that your WAG is not consistent with the SSA table (as you describe its forces of mortality). (The SSA table may not be a good one for your individuals, of course).
It looks to me like you could consider this by summing the probability for year 1,2,…, that the first death occurs that year and that it is the male.
If the male force is double until about age 80, then the conditional probability (at that age) that the male death is first for that couple is 2/3. After male age 80, (at 90 and 87 per your example), it’s more like 1.5/2.5. So the probability across all age at first death would be less than 2/3.
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This method gave me that 47(F) outlives to be .6728 going from current ages to M(119) and F(116). The case where both die in the same year is trivial, only worried about (M) significantly outliving (F).
I also took the calculated t_p_x/ (t_p_x + t_p_y) for the same age ranges , and the average of that is .624, so I believe I have a good estimate.
Thank you so much for your help!