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: First Steps? by TomM)

OK, I've got an idea

First a definition: a number a is of order n where (10^n) ≤ a < (10^[n+1])

If the number of pages in the book a is of order n, then a can be expressed as M(10^n) + D where Mis an integer between 1 and 9, inclusive, and D is an integer less than (10^n)

The page number can also be expressed as m(10^n) + d, but the range for m is between 0 and 9, inclusive.

The probability requested can be expressed as

M
∑r(m) = R
m=0

where r(i) is the probability that m=i [P(i)] times the probability [L(i)] that if m=i, then m lies between 1 and 5, inclusive.

Case 1: 5 ∑ M

For i=0, L(i) = 5/9
For 0 < i ∑ 5, L(i) = 1
For 5 < i ∑ M, L(i) =0
P(i) = 1/(M + 1) (except when i=M)**
r(i) = P(i) * L(i)


Case 2 M ∑ 5

For i=0, L(i) = 5/9
For 0 < i ∑ M, L(i) =0
P(i) = 1/(M + 1) (Except when i=M)**
r(i) = P(i) * L(i)

** When i = M, P(i) is actually [1/(M+1)] * [D/(10^M])

Given any actual number of pages a = M(10^n) + D, it would be relatively easy to plug in the values, but I am too rusty to even attempt solving it as a random distribution. I just hope that this helps in some small way.

Posted by TomM on 2002-07-03 21:23:10