Probability and Independence for Independence Day

The probability I’m talking about probability and independence given that it’s July 4th (aka Independence Day, wink-wink) is 100%.

Yesterday I revisited some probability basics and re-studied what I learned back in college. Instead of explaining everything I learned in depth, I’m just gonna write down the key formulas I read.

So if you ain’t ready for a whole lot of mathematical symbols and letters, then get out of here while you can.

Anyway, here are the topics I’m writing about today:

• Conditional Probability
• Intersection
• Union
• Chain Rule
• Law of Total Probability
• Bayes’ Rule
• Independence (go America)
• Conditional Independence

Conditional Probability

This is the probability an event A occurs given another event B already occurred. This is one of the building blocks for probability and stats. Also what’s cool about this is the way it applies to the real world.

I mean, look at the first sentence of this blog post. It’s pure conditional probability.

Intersection

The intersection of two events A and B is the probability that both A and B will occur.

Union

The union of two events A and B is the probability that either A or B (or both) will occur.

Chain Rule

In calculus, the chain rule relates to differentiating composite functions, which are functions made up of two or more functions.

In probability, the chain rule has a similar purpose.

It allows you to find the probability of multiple intersections by writing the intersections in terms of conditional probabilities.

Law of Total Probability

I got excited when I read this word. While studying stats at Binghamton, the Law of Total Probability was one of the building blocks for the semester.

It gives you to find the probability of event S by considering all possible scenarios and conditions for all A in the same set.

Bayes’ Rule

Ahhh, here’s another old friend from Binghamton and yet another probability building block.

It’s important to know that the following two conditional probabilities are not the same:

• P(A|B)
• P(B|A)

Many mistake them as equal.

Bayes’ Rule is special because it gives you a way to find P(A|B) given you already know P(B|A).

Independence

Now this is what we’re here for! If event A is independent of event B, then the outcome of B has absolutely no effect on the outcome of A.

This statement gives us a useful conditional probability formula to add to our tool belt.

Conditional Independence

Compared to the other topics I wrote about today, I didn’t study this one that much in college.

Conditional Independence allows you to measure whether or not B and C are related given a third condition A.

But it’s important to know that conditional independence does not necessarily mean two events are independent.

Wrapping up

Okay, I’m done. No need for a fancy ending here.

‍