Y is a real number which is chosen at random from the interval (0,1).

Determine the probability that floor(log₁₀(4Y)) = floor(log₁₀Y)

The probablity is 1/6.

First consider only the numbers for which the first non-zero digit is in the 10th-s column.  For the numbers  0.1 <= Y < 0.25, then  .4  <= 4Y < 1 and the condition is met for a probability of 0.15

For numbers for which the first non-zero digit is in the 100th-s column, the condition is met for 0.015  since 0.01 <= Y < 0.025, then  .04  <= 4Y < .1

So the answer is 0.15 times the infinite series 1+a+a^2+a^3+ ... where a=1/10.
And 0.15 * 10/9 = 1/6

Posted by Larry on 2014-08-03 12:18:32