Use the axioms of probability to show the following: (a) For any event E, P(E^')=1-P(E). (b) P(∅

Use the axioms of probability to show the following: (a) For...

Key Concepts

Probability of the Empty Set

The empty set is the event containing no outcomes. By the axioms of probability, since the sample space can be partitioned into the empty set and any event, and since no outcomes contribute probability, the probability of the empty set must be 0.

Subset, Containment, and Monotonicity

This concept involves the idea that if one event is entirely contained within another, then the probability of the smaller event cannot exceed that of the larger event. This follows from the additivity axiom and the non-negativity of probability, since the difference between the probability of the larger event and the smaller event corresponds to the probability of additional outcomes that are present only in the larger event.

Axioms of Probability

The axioms of probability (non-negativity, normalization, and countable additivity) are the fundamental building blocks in probability theory. They allow us to rigorously define how probabilities are assigned to events and to derive properties such as the complement, the probability of the empty set, and the behavior of probabilities under set inclusion.

Complement of an Event

The complement of an event represents all outcomes in the sample space that are not in the event. Using the normalization axiom that the probability of the entire sample space is 1, the complement rule states that the probability of the complement is 1 minus the probability of the event, since the event and its complement are disjoint and together form the entire sample space.

Transcript

00:01 So the axioms of probability, our first one is that the probability of the sample space is 1.

00:10 And then we have between 0, the probability of any event a and 1.

00:16 So any event can be 0 or 1 or between those numbers.

00:21 And then for any event, a1 and a2, for which a1 and a2 is 0.

00:27 So the intersection a1 and a2 is the m -p -set.

00:35 Then this implies that p -a -1 or a -2 is equal to p -a -1, probability of a -1, plus the probability of a -2.

00:45 So those are axioms.

00:52 And then from these, we can go on to say that 1 is equal to the probability of the sample space is equal to the probability of we can call something e or e not.

01:09 So that's going to be, if you imagine a circle, and then this universe, let the circle be e.

01:17 Well, this is e, and then this is e not, which is essentially looking at the whole universe.

01:26 And then what we can say is that this is also equal to the probability of e plus the probability of e complement.

01:36 And this also implies that 1 is equal to the probability of e plus the probability of e complement.

01:46 So that's the first theorem proved, or axiom.

01:50 Now we are going to show, that wasn't a rigorous proof, but it was showing it.

01:55 The probability of the empty set has to be zero.

02:02 So, and we can show it as the probability of the sample space is equal to the probability of sample and union of the empty set, which is going to be the probability of the sample space plus the probability of the empty set...

Please give Ace some feedback