Problem:
Let x be chosen at random from the interval (0,1). What is the probability that
⌊log104x⌋−⌊log10x⌋=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 ⌊log10x⌋=⌊log104x⌋. Thus the condition holds if and only if
n≤log10x<log104x<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