Analysis - How many numbers in the interval [0,1) contain 5 consecutive 5's?

[glacier302]

Homework Statement

Let E be the set of points in [0,1) whose decimal expansion contains somewhere the block 55555. Find the measure of E.

Homework Equations

The Attempt at a Solution

I have a feeling that in order to find the measure of E, I should find the measure of the complement of E and then subtract it from 1. The complement of E would be the points in [0,1) containing no 5's, containing no consecutive 5's, containing 2 consecutive 5's, containing 3 consecutive 5's, containing 3 consecutive 5's, or containing 4 consecutive 5's. I'm not sure how to compute the measure of this set, however.

Any help would be greatly appreciated!

 

[Dick]

Well, start your program. What the measure of all of the points that contain no 5's at all? This is rather like a Cantor set.