Understanding “randomness”

Question

I can\'t get my head around this, which is more random?

rand()

OR:

rand() * rand()

I´m f

Answer 1

Multiplying numbers would end up in a smaller solution range depending on your computer architecture.

If the display of your computer shows 16 digits rand() would be say 0.1234567890123 multiplied by a second rand(), 0.1234567890123, would give 0.0152415 something you'd definitely find fewer solutions if you'd repeat the experiment 10^14 times.

Answer 2

OK, so I will try to add some value to complement others answers by saying that you are creating and using a random number generator.

Random number generators are devices (in a very general sense) that have multiple characteristics which can be modified to fit a purpose. Some of them (from me) are:

• Entropy: as in Shannon Entropy
• Distribution: statistical distribution (poisson, normal, etc.)
• Type: what is the source of the numbers (algorithm, natural event, combination of, etc.) and algorithm applied.
• Efficiency: rapidity or complexity of execution.
• Patterns: periodicity, sequences, runs, etc.
• and probably more...

In most answers here, distribution is the main point of interest, but by mix and matching functions and parameters, you create new ways of generating random numbers which will have different characteristics for some of which the evaluation may not be obvious at first glance.

Answer 3

Oversimplification to illustrate a point.

Assume your random function only outputs 0 or 1.

random() is one of (0,1), but random()*random() is one of (0,0,0,1)

You can clearly see that the chances to get a 0 in the second case are in no way equal to those to get a 1.

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When I first posted this answer I wanted to keep it as short as possible so that a person reading it will understand from a glance the difference between random() and random()*random(), but I can't keep myself from answering the original ad litteram question:

Which is more random?

Being that random(), random()*random(), random()+random(), (random()+1)/2 or any other combination that doesn't lead to a fixed result have the same source of entropy (or the same initial state in the case of pseudorandom generators), the answer would be that they are equally random (The difference is in their distribution). A perfect example we can look at is the game of Craps. The number you get would be random(1,6)+random(1,6) and we all know that getting 7 has the highest chance, but that doesn't mean the outcome of rolling two dice is more or less random than the outcome of rolling one.

Answer 4

I guess both methods are as random although my gutfeel would say that rand() * rand() is less random because it would seed more zeroes. As soon as one rand() is 0, the total becomes 0

Answer 5

Most of these distributions happen because you have to limit or normalize the random number.

We normalize it to be all positive, fit within a range, and even to fit within the constraints of the memory size for the assigned variable type.

In other words, because we have to limit the random call between 0 and X (X being the size limit of our variable) we will have a group of "random" numbers between 0 and X.

Now when you add the random number to another random number the sum will be somewhere between 0 and 2X...this skews the values away from the edge points (the probability of adding two small numbers together and two big numbers together is very small when you have two random numbers over a large range).

Think of the case where you had a number that is close to zero and you add it with another random number it will certainly get bigger and away from 0 (this will be true of large numbers as well as it is unlikely to have two large numbers (numbers close to X) returned by the Random function twice.

Now if you were to setup the random method with negative numbers and positive numbers (spanning equally across the zero axis) this would no longer be the case.

Say for instance RandomReal({-x, x}, 50000, .01) then you would get an even distribution of numbers on the negative a positive side and if you were to add the random numbers together they would maintain their "randomness".

Now I'm not sure what would happen with the Random() * Random() with the negative to positive span...that would be an interesting graph to see...but I have to get back to writing code now. :-P

Answer 6

As others have said, the easy short answer is: No, it is not more random, but it does change the distribution.

Suppose you were playing a dice game. You have some completely fair, random dice. Would the die rolls be "more random" if before each die roll, you first put two dice in a bowl, shook it around, picked one of the dice at random, and then rolled that one? Clearly it would make no difference. If both dice give random numbers, then randomly choosing one of the two dice will make no difference. Either way you'll get a random number between 1 and 6 with even distribution over a sufficient number of rolls.

I suppose in real life such a procedure might be useful if you suspected that the dice might NOT be fair. If, say, the dice are slightly unbalanced so one tends to give 1 more often than 1/6 of the time, and another tends to give 6 unusually often, then randomly choosing between the two would tend to obscure the bias. (Though in this case, 1 and 6 would still come up more than 2, 3, 4, and 5. Well, I guess depending on the nature of the imbalance.)

There are many definitions of randomness. One definition of a random series is that it is a series of numbers produced by a random process. By this definition, if I roll a fair die 5 times and get the numbers 2, 4, 3, 2, 5, that is a random series. If I then roll that same fair die 5 more times and get 1, 1, 1, 1, 1, then that is also a random series.

Several posters have pointed out that random functions on a computer are not truly random but rather pseudo-random, and that if you know the algorithm and the seed they are completely predictable. This is true, but most of the time completely irrelevant. If I shuffle a deck of cards and then turn them over one at a time, this should be a random series. If someone peeks at the cards, the result will be completely predictable, but by most definitions of randomness this will not make it less random. If the series passes statistical tests of randomness, the fact that I peeked at the cards will not change that fact. In practice, if we are gambling large sums of money on your ability to guess the next card, then the fact that you peeked at the cards is highly relevant. If we are using the series to simulate the menu picks of visitors to our web site in order to test the performance of the system, then the fact that you peeked will make no difference at all. (As long as you do not modify the program to take advantage of this knowledge.)

EDIT

I don't think I could my response to the Monty Hall problem into a comment, so I'll update my answer.

For those who didn't read Belisarius link, the gist of it is: A game show contestant is given a choice of 3 doors. Behind one is a valuable prize, behind the others something worthless. He picks door #1. Before revealing whether it is a winner or a loser, the host opens door #3 to reveal that it is a loser. He then gives the contestant the opportunity to switch to door #2. Should the contestant do this or not?

The answer, which offends many people's intuition, is that he should switch. The probability that his original pick was the winner is 1/3, that the other door is the winner is 2/3. My initial intuition, along with that of many other people, is that there would be no gain in switching, that the odds have just been changed to 50:50.

After all, suppose that someone switched on the TV just after the host opened the losing door. That person would see two remaining closed doors. Assuming he knows the nature of the game, he would say that there is a 1/2 chance of each door hiding the prize. How can the odds for the viewer be 1/2 : 1/2 while the odds for the contestant are 1/3 : 2/3 ?

I really had to think about this to beat my intuition into shape. To get a handle on it, understand that when we talk about probabilities in a problem like this, we mean, the probability you assign given the available information. To a member of the crew who put the prize behind, say, door #1, the probability that the prize is behind door #1 is 100% and the probability that it is behind either of the other two doors is zero.

The crew member's odds are different than the contestant's odds because he knows something the contestant doesn't, namely, which door he put the prize behind. Likewise, the contestent's odds are different than the viewer's odds because he knows something that the viewer doesn't, namely, which door he initially picked. This is not irrelevant, because the host's choice of which door to open is not random. He will not open the door the contestant picked, and he will not open the door that hides the prize. If these are the same door, that leaves him two choices. If they are different doors, that leaves only one.

So how do we come up with 1/3 and 2/3 ? When the contestant originally picked a door, he had a 1/3 chance of picking the winner. I think that much is obvious. That means there was a 2/3 chance that one of the other doors is the winner. If the host game him the opportunity to switch without giving any additional information, there would be no gain. Again, this should be obvious. But one way to look at it is to say that there is a 2/3 chance that he would win by switching. But he has 2 alternatives. So each one has only 2/3 divided by 2 = 1/3 chance of being the winner, which is no better than his original pick. Of course we already knew the final result, this just calculates it a different way.

But now the host reveals that one of those two choices is not the winner. So of the 2/3 chance that a door he didn't pick is the winner, he now knows that 1 of the 2 alternatives isn't it. The other might or might not be. So he no longer has 2/3 dividied by 2. He has zero for the open door and 2/3 for the closed door.