Question

I'm having a little trouble with undersanding a spesific line in the proof og the Law of Total Probability (page 79 and 80 in Modern Mathematical Statistics with Applications 2nd edition)

The proof starts with

Because the Ai's are mutually esclusive and exhaustive, if B occurs it must be in cunjunction with exaclty on of the Ai's. That is B=(Ai and B) or .... (Ak and B).

First of all, what does COUNJUNCTION mean in matematics (english is not my first language)

Secone of all, the part "... if B occurs it must be in conjunction with exaclty one of the 'Ai' " doesn't make sense to me. What it B equals the sample space. The it equals the sum som of all Ai's, from A1 to Ak, and not "exaclty one".

Thanks.

Answer 1

The law of total probability is^[1] the proposition that if is a finite or countably infinite partition of a sample space(in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event of   is measurable, then for any event B of the same probability space:

or,

.

So from the definition you have given,there B is an event ,which is in conjuction with means that from the same probability space of ,If B is a event which occurs.

According to the meaning of conjuction in mathematical term .If P is conjuction of Q then it is symbolized as PQ.which in probability term is the intersection of both the events.