Independence

HS #24  2017.7.4

Independence

Any student who takes a course in probability will learn – and often confuse – two important notions: independence and mutual exclusivity. Two events are said to be “mutually exclusive” if they both can’t be true.  One excludes the other. For example, “I am sitting in my lounge chair” and “I am sitting on my couch” are mutually exclusive.

Two things are independent if they have nothing to do with each other – they are totally unrelated. Knowing about one makes the other no more or less likely to be true. For example, “I am sitting in my lounge chair” and “It is snowing on Mount Everest” are independent events. Knowing that I am sitting in my lounge chair reveals nothing about the weather in Tibet, and vice versa.

Other things are neither mutually exclusive nor independent. For example, “I am sitting in my lounge chair” and “It is snowing outside.” They are not mutually exclusive because both could be true at the same time. They are not independent because if it was snowing outside, I would more likely be sitting in my lounge chair than if it was warm and sunny.  That is, the two things are related to some extent.  If one is true, it changes the probability that the other is true.

Try your hand at it. Are the following mutually exclusive, independent, or neither:

i) “I will play racquetball this afternoon.” and “I will spend all afternoon cleaning the house.” ii) “I will play racquetball this afternoon.” and “I didn’t take a shower this morning.” iii) “I will play racquetball this afternoon.” and “You have the TV on as you are reading this.” 

The first is mutually exclusive. The second is neither. (I sometimes wait to take my shower until after a work out.) The third is (most likely) independent.

Notice above that I did not give the option for two things to be BOTH independent AND mutually exclusive. That is because it can’t happen. Do you see why? If I am sitting in my lounge chair, you know for certain that I am not sitting on my couch. If I spend all afternoon cleaning my house, then you know for certain that I did not play racquetball in the afternoon. One being true certainly affects whether the other is true.  So if two things are mutually exclusive they are not independent, and vice versa.

Congrats for your diligent reading. Here (hopefully) is the reward. These two notions help us understand the societal/governmental differences between, say, the United States and France. Also (I think) we can better understand the intentions of our Founding Fathers.

As I understand the laws of France, French citizens are not allowed to wear religious headdresses or other identifying religious garments in the public schools. This is just one example of France’s attempt to keep religion and government (church and state) mutually exclusive. If a person is in school, they are not wearing religious garments. If they are wearing religious garments, they are not in school.

In contrast to that, although they didn’t use the word “independence” (at least in the mathematical sense), I think our Founders had this notion in mind regarding the relationship between religion and government.  They did not want church and state to be mutually exclusive. Instead, their ideal was for them to be independent of each other. One was to have no effect on the other.

In my opinion, this was brilliant. As opposed to France where religion and government have no choice but to push at and exclude each other, in the U.S., the two can coexist with minimal influence on each other.

I say “minimal” because real life is messier than the idealized world of mathematics. If a commencement ceremony of a public school wants to offer a prayer from just one religious point of view and there are objections from those of other religious beliefs, then the government must step in to adjudicate. Nor, unlike businesses and private homeowners, do churches need to pay property taxes. If there were true independence of church and state, churches would pay property taxes like everyone else.

But these are minor. In the grand scheme, our Constitution and legal history are founded on the premise that the church should not officially influence the state, and the state should not affect the church. The wisdom of this approach has led to religions peacefully coexisting together and  thriving unencumbered within our country.  

Independence promotes independence. What a great heritage.