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Imprecise definition of the law of total probability

Imprecise definition of the law of total probability


In the video – lecture 47, it is conveyed that the law of total probability holds when A is the union of some finitely many events B_n for n = 1,2,3…
However, it omitted an important condition that the set of B_n(s) should be a partition of a sample space, in other words, B_n(s) should be mutually exclusive, having no intersections.
Moreover, the video expressed an unnecessary condition: “A =  B_1 U B_2 U … U B_n”. For the law of total probability to hold, A only has to be in the sample space but not necessarily equal to it. That is, “A ⊆ B_1 U B_2 U … U B_n”.
Hope that some amendment could be done to avoid confusion.

1 Answer

365 Team

Hello again, Lal!
For starters, let me clarify that technically you are absolutely spot on. We should have mentioned that the sets B_n are mutually exclusive. However, since we were trying to explain the law with regards to the survey example (and data from surveys as a whole), we though this notion is implied.
Next, since B_1, B_2, … B_n are mutually exclusive, then A being in the sample space is the exact same as A being the sum of some (but not necessarily all B_1, B_2 …). Since we are often only intrigued by the parts that make up our event of interest “A”, it is fair to say that we can take the set of sets (B_m, B_k, etc) that make up A and give them new indexes when talking specifically about A. In this case, we have omitted showing all this background work for simplicity and provided you with “A = B_1 U B_2 U …”, where B_1 represents the “first” part of the sample space for A.
In other words, to avoid over-complicating a seemingly simple enough concept, we just assume A as this new smaller sample space and all the B_i as different parts of it.
To be perfectly honest, we try to make the content as digestible as possible, so we sometimes quickly go over small details such as these for the sake of simplicity. That being said, your observations were completely accurate. We are just trying to explain to you what lead to us presenting the information this way.
The 365 Team

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