Lifespan Probability

Date: 03/26/99
From: Anonymous
Subject: Probability that a general pop will live past 85

My question is:

It is estimated that the probability that the general population will 
live past their 85th birthday is 5.4%. Out of a sample of 600 people, 
what is the probability that fewer than 30 will live beyond their 85th 
birthday? I have tried to use the standard normal distribution to 
approximate but cannot seem to work through it. Can you help?

Date: 03/26/99
From: Doctor Mitteldorf
Subject: Re: Probability that a general pop will live past 85

You can do this problem exactly, without a normal approximation, if 
you use a computer. 

The probability that exactly n people will live past 85 is a product 
of two factors: one is 0.054^n * (1-0.054)^(600-n). This reflects the 
probability that n people will be in the 0.054 group and (600 - n) 
will be in the group with much higher probability (1-0.054). The 
second factor is from Pascal's triangle, and it is the number of ways 
that those n people can be selected from 600: 

   C(600, n) = 600!/(n!(600 - n)!)

With a short computer program, you can add up the terms for 0 through 
29, and find an exact answer = 0.3071.

If you want to use the normal approximation, the formula you need is 
that the mean, quite transparently, is 600*0.054 = 32.4. The standard 
deviation, not so obviously, is sqrt(600*0.054*(1 - 0.054)) = 5.536.  
Now you need either a computer to calculate the area under the normal 
curve, or a table of these values. The table is indexed by z: the 
number of standard deviations from the mean. The question arises, do 
you take (32.4 - 30), (32.4 - 29) or something in between? (The normal 
approximation really only applies to the extent that 30 is a "large 
number" so 1/30 should not matter. But 30 is not such a large number, 
so it does matter.) For the in-between value (32.4 - 29.5), I get 
0.3003 for the probability, which is moderately close to the actual 
value of 0.3071.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/