Today is the Fourth of July and in the US we celebrate Independence Day.

This is a math blog and the author likes puns. Therefore, today not only will we celebrate Independence Day with fireworks, food, and festivities, but we’ll also celebrate independence … the probabilistic kind.

So how does “independence of the probabilistic kind” relate to “independence of the political kind”? Why the word “independence”?

We say that two events \(A\) and \(B\) are independent if \(P(A\) and \(B) = P(A)P(B)\) where \(P(\cdot\)) represents “probability of event \(\cdot\)”.

Here are some examples of independence:

• The probability that the \(nth\) coin flip will land heads given that we have observed \(n-1\) consecutive heads is 0.5. The history of the coin flips does not matter.
• The probability that I will roll snake eyes, when rolling two standard six-sided dice, is \(\frac{1}{36}\). It does not matter if I just observed snake eyes, or if I haven’t observed snakes eyes in the last \(n\) rolls.
• Given a shuffled, standard 52-card deck, the probability that the first card you draw is the ace of spades is independent of where from the deck you draw the card. The ace of spades is just as likely to be at the top of the deck as it is to be at the bottom of the deck as it is to be in any other position in the deck.

In probability theory one can think of independence as a way to state that two events are unrelated, uncorrelated. That is, no matter what happens in that other event, this event, the event we care about, is unaffected and vice versa.

Similarly, this is the bold statement of political independence. It is a statement that no matter what that other country does, our country will not be (negatively) affected by it. Our fate is independent of the goings-on elsewhere. When we declared independence from Great Britain, we were making that bold statement. We were saying that we decide our fate. We decide our own laws and that the laws of Great Britain do not govern us — that we are not dependent on the British legal system. This is the independence that we declared.

Thus, if we think about independence from the standpoint of “law”, be it legal, physical, etc. then we can understand how coin flips and political independence are similar in concept. For example, what is the rule governing a coin flip? They are physical rules; things like gravity, the laws of motion, etc. So ask yourself, how the fate of the next coin flip governed by the fate of this coin flip? Does the coin inherently remember its previous flips? Is it affected by its previous flips? By all practical measures, a coin has no memory of its previous states and thus it just “lives in the moment”.

Before the persnicketiness sets in, there are three type of independence: “\(A\) is independent of \(B\)”, “\(B\) is independent of \(A\)”, and “\(A\) and \(B\) are independent of each other”. Political independence usually has the connotation of “one-way” independence. Namely, \(P(A | B) = P(A)\) means that \(A\) is independent of \(B\); whatever the result of \(B\) is, it does not change the likelihood of occurrence of \(A\).

Just a light thought for a holiday. Happy Fourth of July!