Are you looking for the range of values x for which A(B^-x) < (B-x)! holds good? If so then the range is
x <= -32
x >= 32
for values of x between -31 to 31 (included) it is invalid. Did it iteratively.
Thanks for your effort.
I'd actually already solved it iteratively using a calculator, but the people setting the question wanted a more analytic solution
The trick is to use Stirling's (simple) formula, ln (x!) approximately equal to x.ln (x) - x
I still didn't get a solution before I ran out of time.
the range you came up with though was correct, which at least shows I was on the right track (I think!)
blimey! I havn't used iterative processes for years... I'm an undergrad physicist and
If it helps, the problem is thus:
What is the minimum number of people you need in a room for there to be a 75% chance that two people share the same birthday.
my workings:
P(n people do NOT share the same birthday) = 1 x (1- 1/365) x (1 - 2/365) x ... x ( 1 - n/365)
here the first factor is the first person entering the room, etc
we are looking for n such that
PRODUCT'SUM' from i=0 to i = n-1
(1 - i/365) = 0.25 *
LHS = PRODUCT'SUM' 365^n (365 - i)
= 365 ^ n [365 ! / n !]
which, fitting this all into the * equation, gives us
365 ! / 0.25 = 365^-n * n!
hmm, perhaps I made an error when I last did it... anyway this was where I got stuck.
though do remember that I have already handed my solutions in, I would still like to see the solution
If you are trying to solve it, as it is, I suggest you not to use Stirling's formula. but from Stirling's you get .
Another way to find n is to use Taylor series expansion for the probabilities as each person enters the room.
Substitute,
.
Probability of n people not sharing birthday becomes,
which can be further reduced since,
Now,
Since,
Using simplify,
For , you get
Hope this helps.