What Brilliant subscription is more convenient to me? Statistics problem
I am not affiliated to Brilliant, just found it a fun problem to solve
To calculate what subscription to Brilliant is more convenient to me, I collected some data.
My life expectancy at 50 in my country (Italy) is 82,1 years + I gave myself another 5 years for good health condition and reasonable well being (and beacuse I really wish I had them...). But I removed 5 years for I imagined not being in perfect brain shape for the last 5 yeas of my life (far from brilliant, let's say).
That gave me 37,1 years of brilliant life expectation (geez D: got to hurry up some stuff), performing all calculations, just like this, the lifetime subscription costs about 1.17/month.
But then of course not all users stay for all their life, sometimes they get annoyed, change interests, sites go out of business (sorry Brilliant :D LOL) end up all the site content ( :D ) etc. so I considered a dropout rate for whatever the reason be it.
I define a dropout rate d as the percentage of people that stop renewing every year. That is the same as considering the probability chance that one single person stops renewing that year. I tried two options, a dropout rate of 20% (every year I have a 20% chance of getting tired and stopping using Brilliant), and a dropout rate of 10% a year.
What is the most convenient option at a droput rate of 10% at my age in my country?
Do not take into account:
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that the lifetime subscription grants you the use of the service even if you would have otherwise dropped out (it does have a percieved value, that is less than the subscription price, but still not zero)
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that you would stop renewing the monthly fee much sooner than the yearly, being more expensive, just keep the 10% dropout rate
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that the monthly subscripion has sense to test the service or follow a given course of interest if you do not plan on long term loyalty
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that a lot of money instantly is not the same as the same amout in a long time frame or in the future (like a loan is) so 5.5/month is probably easier on your pockets than 450 now.
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just count the cash, forget all the rest
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1 solution
The curves of the "remaining" (s for "stay") chance at time t (in years) for those options are
s(t) = 0.8^t {t>0} at 20% droput rate
s(t) = 0.9^t {t>0} at 10% droput rate
l(t) = 1 - s(t) is the probability of having left that year or before
[every year I have the (1-d)% of chance to stay. At year n, (1-d)^n stays, so the probability of going away at year n is 1-(1-p)^n]
When l(t) = s(t) or
s(t) = 0.5
we have half the people has gone away, so we have the mean life on the site, or 50% chance of unsubscribing.
This happens at 0.9^t= 1/2 or t = log0.9( 1/2 ) or t = ln( 1/2 ) / ln(0.9) = 6.57881 years. That is far less than my 37,1 years of brilliant life expectation. For that number of years...
So apparently the yearly option is the most convenient (but on the lifetime you will keep brilliant).