Why does the same proportion of a radioactive substance decay per time period? (half life) Just wondering, if decay is random, why does the activity half every half life, as in, why does it have to reduce by the same proportion in the same time period?
 A: You're asking the wrong question.
There is no magical notion by which it decays by half.  That is why "half-lives" vary so much.  "Half-lives" are simply the chosen method of measurement.
Your question is analogous to asking "why all cars travel in hourly increments?" just because we measure speed in km/h.
The notion implicit is using half-lives, is that given a large enough sample (1 mol = 6 E23) the rate of decay is close enough to constant.  ie:  That if say in any second, the "chance" of decay is X% for any atom, then over such a huge sample X will present as a constant.  For example, if we said that a person having open-heart surgery has a 0.1% chance of dying on the table, we would not expect that to be accurate over a small sample.  We couldn't say "well only 10 of us are having the operation today, so I'm safe."  But over many trillions of such operations, we'd expect the 0.1% to hold true.
In summary, a half-life is simply a different way of presenting a constant rate of decay.
(Keeping in mind that a constant RATE of decay, delivers progressively smaller decay as the amount of un-decayed material reduces.)
A: The word random in this context does not mean totally without order.
What it means is that one cannot predict exactly when a particular unstable nucleus will decay although there is an underlying probability of decay of an unstable nucleus in a specified interval of time.  
An interval of time which is often used for a particular species of unstable nucleus is the half-life.
The probability that an unstable nucleus will decay in a time interval equal to one half life is $\frac 12$.
If an unstable nucleus does not decay in that time interval then the probability that it will decay during the next time interval of the same length is still $\frac 12$ . . . etc.  
You will note that this is similar to the toss of a coin with two outcomes heads and tails each with a probability of $\frac 12$.
However an insight to the randomness of coin tossing might be shown in that although the probability of tossing a head is $\frac 12$, then if one tossed a coin $100$ times there is a fairly low probability, $0.07959$, of the outcome being exactly $50$ heads and $50$ tails.
So what you have is a number of possible outcomes, number of heads + number of tails $=100$, for which you can predict the probability of them happening but you cannot say for certain, probability $=1$, which of those outcomes will actually occur.  
In the context of radioactive decay on average half a sample of unstable nuclei will decay in one half life and then on average half the remaining unstable nuclei will decay during the next interval of one half life etc.
With samples of billions and billions of unstable nuclei the statistical fluctuations about “one half will decay during a time interval of one half life” will be small.
As the number of unstable nuclei becomes smaller the statistical fluctuation about one half will get larger.
Just think about what you would predict about the decay of $3$ unstable nuclei in an interval of one half life. They could all not decay for ten half lives although the probability of the happening is small. 
A: An example that might help:
Start with a big pile of coins. Flip them. Remove the heads. About half remain. 
Take the remainder and flip them. Remove the heads. About half remain. 
Take the remainder and flip them. Remove the heads. About half remain. 
The analogy: An atom has a 50% chance of decaying in some particular interval $T_{1/2}$. After each of those intervals, half are left. 
A: It's a consequence of the fact that the nucleus doesn't know how many other nuclei are in your lump of material. A 3kg block of uranium has to decay at the same rate as three 1kg blocks of uranium. Which means a 1kg block has to decay at 1/3 of that rate. A 1kg block is the same as 3 1/3kg blocks, so a 1/3kg block has to decay at 1/3 of that rate too.
Now suppose you have a 3kg block of uranium (or whatever-ium), and it takes a year to decay into 1kg of whatever-ium (and 2kg of other stuff - let's pretend you have a system to take that away because we're only talking about the uranium here). Since you have 1kg, it must decay at 1/3 of the rate that it did at the start. It takes the same time to decay 2/3kg from a 1kg block as it does to decay 2kg from a 3kg block. That means after a year, you only have 1/3kg left. And decaying 2/9kg from a 1/3kg block takes the same time as decaying 2/3kg from a 1kg block. So after another year, you have 1/9kg left. And so on.
We say that whatever-ium has a third-life of one year.
We can extrapolate with maths. We know it has a ninth-life (1/3 squared) of two years. We know it has a 57.3%-life (square root of 1/3) of half a year. We know it has a half-life of 0.63092975357 years (you need to use logarithms to work this out).
We measure things in half-lives because it's convenient. We could equally well use third-lives or quarter-lives or fifth-lives or two-thirds-lives.
A: A couple of answers above hit it well. Here is a slightly different perspective.
From an visual standpoint, consider a pointillist painting. If you look at any single dot up close, the painting makes no sense. Stand back, and order falls into place.
The term “random” does not mean without order. It means that nothing we know up to this point with this particular perspective enables us to predict its function going forward in time. 
