Bascule is reading a book.
What is the probability that the first digit of the page he is on is 1, 2, 3, 4 or 5?
a) obtain an expression
b) approximate a numerical value
(In reply to
re(3): well.. by friedlinguini)
the way you have defined g(n), it yields the probability of
a page starting with [1,5] if the book has n pages.
eg g(9) = 1/10 + 1/10 + 1/10 + 1/10 + 0/10
+ 0/10 + 0/10 + 0/10 + 0/10
= 5/9
this g(n) however is not the function i have in mind, mainly
because it is defined in terms of the exact number of pages of the
book. consider a "higher order" function r(n). i say higher order
because the number of pages n is itself a random variable, subject
to probability. in this way r(n) is a function of ranges not values.
for example r(1-9) which we can for convenience call r(1):
r(1) = 5/9 * 1 + 1/9 * 5/6 + 1/9 * 5/7 + 1/9 * 5/8 + 1/9 * 5/9.
(jim lyons post has something like this)
in other words r(n) incorporates the likelihood of a certain number
of pages in the probabilities in order to encompass
ranges.
let me know if this makes sense before i continue..