How do i know if this is random enough?

Question

I wrote a program in java that rolls a die and records the total number of times each value 1-6 is rolled. I rolled 6 Million times. Here's the distribution:

#of 0's: 0
#of 1's: 1000068
#of 2's: 999375
#of 3's: 999525
#of 4's: 1001486
#of 5's: 1000059
#of 6's: 999487

(0 wasn't an option.)

Is this distribution consistant with random dice rolls? What objective statistical tests might confirm that the dice rolls are indeed random enough?

EDIT: questions have been raised over application: a game that i want to be as fair as can be reasonably achieved.

Answer 1

If your random number generator passes the Diehard tests, that's the best you can do.

Even a physical die won't be perfect with 1/6 per face.

Increase the trials by an order of magnitude, then do it again. If you get 1/6 for each trial you'll be fine.

Answer 2

This test alone isn't enough to determine randomness. Not that it's completely useless, but a "random" dice roller that outputs 1,2,3,4,5,6 and repeats would be perfectly random according to this test.

Another suggested test: pick a number, x, and each time it is rolled, record the statistics of what number comes next; you should see an even distribution again. Repeat for all six values of x. If it passes this test it is probably random enough to be used as a dice roller.

Answer 3

To test whether this particular distribution is consistent with the expected distribution of numbers rolled with a "fair" dive, you need to perform the Pearson's Chi-square test.

Note that this still will not prove that your algorithm is "fair", only that these particular results look "fair".

To test whether your algorithm is "fair" in general, use the Diehard tests, as others have mentioned.

Answer 4

The probability that 6'000'000 dice rolls will end up in exactly 1'000'000 outcomes of each is close to 0. As long as the sum if the outcomes is correct, and that the variance (error) of the difference from the expected outcome goes towards 0 (relatively) when the number of trials increase, then your random function is not wrong.

You can either prove it mathematically or by testing the random function with larger and larger trial sequences to see that it converges.

For a repeated number of tests, the sum for each outcome should approximate Gaussian distribution. E.g. each outcome 1-6 should fall within normal distribution centered around 1'000'000 with a variance that is inversely proportional to the number of dice rolls.

The other tests, the Diehard tests, tests that the actual sequence of dice rolls is random in itself and not that the outcome of 6'000'000 rolls for example is 100'000 consecutive 1's, then 100'000 2's and so on and finally some random sequences.