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Random log (Posted on 2018-02-04) Difficulty: 3 of 5
If r is a uniformly distributed random number between 0 and 1, what is the probability that floor(log2(1/r)) is odd?

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Solution Solution with generalization Comment 1 of 1

From 1/2 to 1 the function is 0: even
From 1/4 to 1/2 the function is 1: odd
From 1/8 to 1/4 the function is 2: even
From 1/16 to 1/8 the function is 3: odd
Each even interval has a corresponding odd interval which is half as wide, so overall the P(even)=2*P(odd) 
P(even) = 2/3
P(odd) = 1/3
This can also be seen as an alternating geometric series
1/2-1/4+1/8-1/16+1/32-1/64 ... = 1/3

In general for base b logarithm
P(odd) = 1/(b+1)

The above formula works for any positive real base (except 1) 

  Posted by Jer on 2018-02-04 13:19:07
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