# Question on Actuary Exam -- Error?

• I
Gold Member
Hi All,
I was trying to help someone in a question for the actuary exam. I wonder if there is a mistake or whether I am missing something obvious here:

What is the age a 90-year old man can expect to live given the probability an x year old man will live until age x+1 is given by ## P(x+1|x )= \frac {x-89}{5} ##? We then compute p(91|90)90+.....+p(94|93)94. But we don't have an answer sheet.

I assumed from the terms that the person is not expected to live beyond 94. Is this the way to do it or am I missing something

## Answers and Replies

DrClaude
Mentor
This is definitely not my field, but there are a few things I find strange in what you wrote.

First, shouldn't the expected age be of the form 90 + something, since you are assuming the man is already 90? In any case, I find it strange to see ##P(91|90)## multiplying 90 instead of 91.

Also, shouldn't you have ##P(x+2|x) = P(x+2|x+1) P(x+1|x)## and so on?

WWGD
Gold Member
This is definitely not my field, but there are a few things I find strange in what you wrote.

First, shouldn't the expected age be of the form 90 + something, since you are assuming the man is already 90? In any case, I find it strange to see ##P(91|90)## multiplying 90 instead of 91.

Also, shouldn't you have ##P(x+2|x) = P(x+2|x+1) P(x+1|x)## and so on?
Yes, those are a few among many things that are unusual about the question.

DrClaude
Mentor
Yes, those are a few among many things that are unusual about the question.
I think it would be useful to restate the problem using the homework template (actually, this probably belongs in the homework forum).

Gold Member
I think it would be useful to restate the problem using the homework template (actually, this probably belongs in the homework forum).
This is not a HW problem. This is a practice problem for Actuary exams from a few years back. I know the material, it is just that the layout seems unusual to me.EDIT: I am just asking for some clarification, not an answer.

mathman
Hi All,
I was trying to help someone in a question for the actuary exam. I wonder if there is a mistake or whether I am missing something obvious here:

What is the age a 90-year old man can expect to live given the probability an x year old man will live until age x+1 is given by ## P(x+1|x )= \frac {x-89}{5} ##? We then compute p(91|90)90+.....+p(94|93)94. But we don't have an answer sheet.

I assumed from the terms that the person is not expected to live beyond 94. Is this the way to do it or am I missing something
The formula looks strange. P(91|90)=1/5 while P(92|91)=2/5 and P(95|94)=1. I suspect the formula is for probability of death?

StoneTemplePython
Gold Member
Hi All,
I was trying to help someone in a question for the actuary exam. I wonder if there is a mistake or whether I am missing something obvious here:

What is the age a 90-year old man can expect to live given the probability an x year old man will live until age x+1 is given by ## P(x+1|x )= \frac {x-89}{5} ##? We then compute p(91|90)90+.....+p(94|93)94. But we don't have an answer sheet.

I assumed from the terms that the person is not expected to live beyond 94. Is this the way to do it or am I missing something
It seems strange to have ##(x-89)## in there. I.e. this transition probability doesn't make any sense for an 88 year old. It also doesn't make sense for 95 year old. Also consider that ##\frac{1}{5} + \frac{1}{5}\frac{2}{5} + \frac{1}{5}\frac{2}{5}\frac{3}{5} + \frac{1}{5}\frac{2}{5}\frac{3}{5}\frac{4}{5} + + \frac{1}{5}\frac{2}{5}\frac{3}{5}\frac{4}{5}\frac{5}{5} \lt 1## so your interpretation implies that there is non-zero probability of the guy living beyond 94, which raises a contradiction because the transition probability of a 95 years old is greater than one.

I have 2 guesses:

guess 1: you are supposed to have a transition probability tailor made for each year of age and for some reason you don't have this.

guess 2: be overly literal and say ##p=\frac{90-89}{5}= \frac{1}{5}## and treat this as a geometric series that indexes at 0 (as opposed to the customary 1) which gives the expected value of ##\frac{1}{\frac{1}{5}} -1 = 4##, with our 'origin' at 90, so the expected age of 'absorption' is 94.

MarneMath
Typically actuarial exam question. Given that someone is 90 the probability they live until 91 is .8. Probability someone who makes it to 92 is .48. .8 -.48 = .32 so .32 die before reaching 92. Repeat this procedure over and over again and then sum each term with respect to the age, you'll end up with 91.5 which is the currently answer.

(I was an actuary in a past life)

WWGD
Gold Member
Typically actuarial exam question. Given that someone is 90 the probability they live until 91 is .8. Probability someone who makes it to 92 is .48. .8 -.48 = .32 so .32 die before reaching 92. Repeat this procedure over and over again and then sum each term with respect to the age, you'll end up with 91.5 which is the currently answer.

(I was an actuary in a past life)
Thanks. So from this it follows that living behind 95 under these conditions is not an option (Y/N; please, no answer)? Besides EDIT: P(91|90)=(90-89)/5 so 1-P(91|90)=0.2 .

Last edited:
mfb
Mentor
I agree with mathman, this looks like the probability of death instead of survival, otherwise 84 year olds are immortal. If it is the probability of death then no one can reach 85 and it makes sense to stop the sum after 84.
There is a 1-P(91|90)=0.2 probability that the person dies at 90.
There is a (1-P(92|91))P(91|90) = 0.4*0.8 probability that the person dies at 91.
And so on.
Multiply the probabilities by the ages and you get the life expectancy.

WWGD