Question on Actuary Exam -- Error?

In summary, a 90-year old man has a probability of 0.2 of dying at age 90, 0.32 of dying at age 91, 0.24 of dying at age 92, 0.12 of dying at age 93, and 0.04 of dying at age 94. Therefore, the expected age a 90-year old man can live is 91.5 years. The formula given for the probability of survival may actually be for the probability of death instead.
  • #1
WWGD
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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
 
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  • #2
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?
 
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  • #3
DrClaude said:
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.
 
  • #4
WWGD said:
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).
 
  • #5
DrClaude said:
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.
 
  • #6
WWGD said:
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?
 
  • #7
WWGD said:
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.
 
  • #8
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)
 
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  • #9
MarneMath said:
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 .
 
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  • #10
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.
 
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1. What causes errors on actuary exams?

Errors on actuary exams can be caused by a variety of factors, such as misinterpretation of the problem, calculation mistakes, or unfamiliarity with the material. They can also be caused by test-taking strategies, such as rushing or not reading the question carefully.

2. How can I avoid making errors on actuary exams?

To avoid errors on actuary exams, it is important to thoroughly understand the material and practice solving problems before the exam. It is also helpful to carefully read and understand each question before attempting to answer it. Taking breaks during the exam to clear your mind can also help prevent errors.

3. Can I request a regrade if I believe an error was made?

Typically, actuary exams have a strict no regrade policy, so it is important to double check your work and make sure you understand the material before taking the exam. However, if you believe there was an error in grading, you can reach out to the exam administrator for clarification.

4. How do errors affect my overall score on an actuary exam?

Errors can significantly impact your overall score on an actuary exam, as they can result in incorrect answers and deductions of points. It is important to minimize errors to maximize your chances of passing the exam.

5. What can I do if I consistently make errors on actuary exams?

If you consistently make errors on actuary exams, it may be helpful to seek additional resources such as study guides or tutoring to improve your understanding of the material. It may also be helpful to identify any common patterns in your errors and work on addressing them specifically.

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