Fairness

I've personally said something similar to the following (without much thinking of its scientific meaning, though):

"He's good at academics, sports and music, but he's short and ugly. God is fair - usually people cannot have all the good things."

Now I'm going to consider this "fairness" from a probabilistic perspective (and let's put the theological issues aside - I will assume God exists). In probability, "fair" has a special meaning. A fair coin is a coin that will equally likely give head or tail in every toss, which is independent of all previous tosses. (That is, even when the previous 10 tosses are all heads, there is still a 50% chance of getting a head in the 11th toss.)

If one tosses 3 fair coins sequentially, one has an 1/8 chance of getting each of the following eight sequences (H denotes heads and T denotes tails):
1. HHH (3H)
2. HHT (2H1T)
3. HTH (2H1T)
4. HTT (1H2T)
5. THH (2H1T)
6. THT (1H2T)
7. TTH (1H2T)
8. TTT (3T)

(The statistics in the parentheses give the total number of heads and tails. For example, "2H1T" means 2 heads and 1 tail in total. It happens in sequences 2, 3 and 5.)

Suppose your friend has made the coin tosses and told you that any two of the three coins are heads (i.e. the 3 coins are HHX, HXH or XHH. You don't know which two coins he is referring to). Do you think the unknown coin X is more likely to be head or tail?

The correct answer is tail. We can see that your friend is referring to sequences 1, 2, 3 or 5, and only sequence 1 has X equal to H. Since all sequences are equally likely, based on what your friend tells you, the unknown coin has a 75% chance of being tail and only 25% chance of being head.

Let's translate all these into the context of the example: suppose the first coin governs your "academic" aptitude, the second coin refers to "sports and music," and the third coin determines your "appearance." Assume head means "good" and tail means "bad."

My argument is valid if I, like your friend, start with any two unspecified attributes (coins). I observe that a guy has two good attributes (two heads), so I conclude that his remaining attribute is more likely to be bad (tail). This is correct in terms of fairness.

However, if I observe the first two attributes (i.e. they are specified) and see that the guy is good at academics, sports and music, then what I said is not logically sound. This is because being "short and ugly" now has nothing to do with his academic and sports and music abilities. In the coin toss case, if your friend tells you that the first two coins are heads, i.e. the coins are HHX (it's either sequence 1 or sequence 2), whether the third coin X is head or tail is 50-50, and is independent of what you are told, since the coin is fair.

This example highlights the different meanings of "fairness" in different situations. Even from a statistician's point of view, fairness does not always mean 50-50. Before making claims such as "God is fair," we should clearly state what our assumptions are.