Calculating the Min Entropy of a string of bits

Let's say you are looking to collect some randomness from your computer. At a given time stamp, you can always collect the following 16bits as part of the time stamp:

• First 5 bits are fixed and always the same
• The next 4 bits are not perfectly random and exhibit a probability of 1/8 for the different possibilities
• Finally the last 7 bits are perfectly random, always

What would be the min entropy of one time stamp?

My intuition thinks the min entropy would be 3 bits (from the 4 imperfect bits) + 7bits from the last section, giving Min Entropy = 10 bits for one time stamp. Does this make sense?

EDIT: Rephrased the 2nd bullet point

• Your second sentence makes no sense. There are 16 possible combinations, they can't all have the probability 1/8, since that's 200%. Also that'd be uniform "perfectly random". – CodesInChaos Oct 25 '16 at 18:15
• You're right, let me rephrase it. – guy Oct 25 '16 at 18:17
• I still don't get your second point. You mean that 8 of the 16 possible values are chosen with probability 1/8 and the other 8 possible values are never chosen? – CodesInChaos Oct 25 '16 at 18:20
• Yes exactly! Let me know if I should rephrase it again. I understand it is strange. (Hence my confusion) – guy Oct 25 '16 at 18:23
• The math sounds about right to me. – SEJPM Oct 25 '16 at 20:29

If your probabilities are exact, then your math is right. But even if they are off by the slightest amount (e.g. $2^{-10000}$), it doesn't hold any more. For a practical system like timestamps in a computer, it's quite unlikely that your have exact values for the probabilities. If those probabilities are average measurements, then the resulting entropy is also only an average and not a lower bound (minimum).