# Does God play dice with the universe?

Imagine I gave you an ordinary die, not special in any way, and I asked you to throw the die and record your results (how many 1s, how many 2s, etc.). What would you expect the results to be? Do you think you could win by choosing some numbers rather than others? Are you sure?

(Image source: Wikimedia Commons. Author: Diacritica. License: Creative Commons.)

# What you might expect

Let's say you thew the die 12,000 times, you might expect a probability distribution something like this. This is a uniform distribution where all results are equally likely.

You know you'll never get an absolutely perfect distribution, so in reality, your results might look something like this for 12,000 throws.

The deviations from the expected values are random noise which we can quantify. Further, we know that by adding more dice throws, random noise gets less and less and we approach the ideal uniform distribution more closely.

I've simulated dice throws in the plots below, the top chart is 12,000 throws and the chart on the bottom is 120,000 throws. The blue bars represent the actual results, the black circle represents the expected value, and the black line is the 95% confidence interval. Note how the results for 120,000 throws are closer to the ideal than the results from 12,000 throws.

# What happened in reality - not what you expect

*more frequently*than theory would suggest - the deviation from the uniform distribution is

*statistically significant.*The dice he used were not special dice, they were off-the-shelf standard unbiased dice. What's going on?

# Unbiased dice are biased

*indented*. Each hole is the same size, but obviously, the number of holes on each face is different. Let's think of this in terms of weight. Imagine we could weigh each face of the dice. Let's pair up the faces, each side is paired with the face opposite it. Now let's weigh the faces and compare them.

# Vegas here I come - or not...

*not*indented. They're filled with colored material that's the same density as the rest of the dice. In other words, there's no imbalance, Vegas dice will give a uniform distribution, and 1 and 6 will occur as often as 2, 3, 4, or 5. You're going to have to keep punching the clock.

# Some theory

- \(n\) is the number of throws
- \(p\) is the probability of getting a 1, which is \(\frac{1}{6}\)
- \(q = 1- p\) is the probability of getting 2-6, which is \(\frac{5}{6}\)

- The mean is \(np = 12 \times 26,306 \times \frac{1}{6} = 52,612\)
- The standard deviation is \(\sqrt{npq} = \sqrt{12 \times 26,306 \times \frac{1}{6} \times \frac{5}{6}} = 209.388\)
- The 95% confidence interval is \(52,202 \) to \(53,022\) (standard deviation by 1.96).