Problem:
Let x be chosen at random from the interval (0,1). What is the probability that
⌊log10​4x⌋−⌊log10​x⌋=0?
Here ⌊x⌋ denotes the greatest integer that is less than or equal to x.
Answer Choices:
A. 81​
B. 203​
C. 61​
D. 51​
E. 41​
Solution:
The given condition is equivalent to ⌊log10​x⌋=⌊log10​4x⌋. Thus the condition holds if and only if
n≤log10​x<log10​4x<n+1
for some negative integer n. Equivalently,
10n≤x<4x<10n+1
This inequality is true if and only if
10n≤x<410n+1​
Hence in each interval [10n,10n+1), the given condition holds with probability
10n+1−10n(10n+1/4)−10n​=10n(10−1)10n((10/4)−1)​=61​​
Because each number in (0,1) belongs to a unique interval [10n,10n+1) and the probability is the same on each interval, the required probability is also 1/6​.
The problems on this page are the property of the MAA's American Mathematics Competitions