I'm trying to calculate the probability, and potential cost on society, of people returning to homelessness after going through the system one, two, or several more times.
Let's say that someone who enters into a homeless shelter and eventually finds suitable housing ends up costing the homeless assistance system $\$2000$ to provide all the services that helped them to resolve their housing crisis. There is a 35% chance that this person will, for any of a whole slew of reasons, return to homelessness after 2 years. In other words, after $\$2000$ was spent on them to find them a place to live, there's a 35% chance that within two years they'll end up homeless again and have to enter into a shelter, go through the whole process, and generate another $\$2000$ of costs to find them a place to live.
My question is, since this situation is recursive, in that a person could potentially become homeless over and over again throughout their lifetime, how do I calculate the "average scenario", or the cost that any given homeless person will generate on the system, over their lifetime, given the information above?
The tricky part for me is figuring out what to do with the 35%, since it's the chance that they'll return to homelessness after 2 years, not over their entire lifetime.
Any ideas?
Okay so for the first iteration you have:
Note: where X is the number of homeless.
$X*2000$
This is before any one has a change to be homeless again.
If you include the second iteration you have:
$X*2000 + X*2000*0.35$
If you include the third iteration you have:
$X*2000 + X*2000*0.35 + X*2000*0.35*0.35$
Written as a summation:
$\sum_{i=0}^n X*2000*0.35^i$
Where n is the number of times you want to iterate the process. It is important to note that this is ignoring any background rise or drop in the homeless
If you calculate it as n goes to $\infty$ you would get your answer (currently I cannot remember how to do this). But for n=10 you get $3076.84.
Please note I'm not being very exact as the 35% is for within 2 years so n=10 does not mean the average cost of the person over 20 years as they could become homeless in less than the 2 years.