Axioms of Probability

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Cuthbert Perkins
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1 Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing two dice, then the sample space consists of the 36 points. Solution:

2 Event Any subset of the sample space is known as an Event, and usually denoted by the letters A, B, C, etc. If the outcome of the experiment is contained in an event E, we say the event E has occurred. In general, we are interested in finding the probability that a given event has occurred. Example 1.11

3 Union of the Events E and F (denoted by E F) For any two events E and F from the sample space S, the event C is called the union of the events E and F if C consists of all outcomes that are either in E or in F, or in both E and F, denoted by C = E F. Note 1.5: The event E F will occur if either E or F occurs. Example 1.12 Note 1.6: If E 1,E 2,..., are events from the sample space S, then the union of the events n=1 E n is the events that consists of all outcomes that are in E n for at least one value of n = 1,2,...

4 Intersection of Events E and F (denoted by EF or E F) For any two events E and F from the sample space S, the event C is called the intersection of the events E and F if C consists of all outcomes that are both in E and in F, denoted by C = E F. Note 1.7: The event E F will occur only if both E and F occurs. Example 1.13 Note 1.8: If E 1,E 2,..., are events from the sample space S, then the intersection of the events n=1 E n is the events that consists of outcomes which are in all of the event E n, n = 1,2,...

5 Null Event (denoted by ) If the event does not contain any sample point, we refer it as the null event, and denoted by. Mutually Exclusive Events If E F =, the events E and F are called mutually exclusive events. Example 1.14

6 Complement of E (denoted by E c ) The complement of the event E consists of all outcomes in the sample space S that are not in E, and is denoted by E c. Note 1.9: E c will occur if and only if E does not occur. For example, Note 1.10: S c = and c = S.

7 E contained in F (denoted by E F or F E) For any two events E and F in the sample space S, if all of the outcomes in E are also in F, then the event E is said to be contained in the event F, and denoted by E F or F E. Note 1.11: If E F, then the occurrence of E implies the occurrence of F. Note 1.13: If E F and F E, then E and F are equal, and written by E = F.

8 Venn Diagrams The Venn Diagram is a graphical representation for illustrating logical relations among events. The sample space S is represented as a larger rectangle. Events are represented as circles within that rectangle. Events of interest can then be indicated by shading appropriate regions of the diagram. Example 1.15 Given the events E, F and G from the sample space S, indicate the following events in the Venn diagrams: (1) E F, (2) E F, (3) E c, (4) E F, (5) E F =, and (6) (E F ) G. Solution:

9 Rules of Operation The operations of forming unions, intersections, and complements of events obey certain rules. Commutative Laws E F = F E and E F = F E. Associative Laws ( ) ( ) E F G = E F G and ( E F ) G = E ( F G ). Distributive Laws ( E F ) G = ( E G ) ( F G ) and ( E F ) G = ( E G ) ( F G ).

10 DeMorgan s Laws and Proof: (When n = 2) ( n ) c n E i = Ei c i=1 i=1 ( n ) c n E i = Ei c. i=1 i=1

11 Suppose an experiment with sample space S is repeatedly performed under exactly the sample conditions. For each event E of the sample space S, we define n(e) to be the number of time that the event E occurs in the first n repetitions. The probability of the event E, denoted by P(E), is defined to be P(E) = lim n n(e) n. For example, suppose we repeatedly toss a coin, how do we know that the proportion of heads obtained in the first n flips will converge to some value as n gets large?

12 Modern Axiomatic Approach to Probability Theory For each event E in the sample space S, there exist a value P(E), referred to as the probability of E. All these probabilities satisfy a certain set of axioms. Axioms of Probability Axiom 1 For any given event E, 0 P(E) 1. Axiom 2 P(S) = 1. Axiom 3 If E 1,E 2,... is a sequence of mutually exclusive events, then P(E 1 E 2 ) = P(E 1 ) + P(E 2 ) +.

13 Note 1.14: The null set has probability 0 of occurring. That is Proof: P( ) = 0. Note 1.15: For any limit sequence of mutually exclusive events E 1,...,E n, Proof: P(E 1 E 2 E n ) = P(E 1 ) + + P(E n ).

14 Example 1.16

15 Some Simple Propositions Proposition 1.1 Given the event E in a sample space S and E c is the complement of E, we have Proof: P(E c ) = 1 P(E). Example 1.17 From Example 1.16, what is P { Rolling an odd number }? Solution:

16 Proposition 1.2 Given the events E and F in a sample space S and E F, we have Proof: P(E) P(F ). Proposition 1.3 If E and F are two events in a sample space S, then Proof: P ( E F ) = P(E) + P(F ) P(E F ).

17 Proposition 1.4 If E, F and G are three events in a sample space S, then Proof: P ( E F G ) = P(E) + P(F ) + P(G) P(E F ) P(E G) P(F G) + P(E F G).

18 Proposition 1.5 If E 1,...,E n are n events in a sample space S, then P ( E 1 E n ) Proof: (by induction) = n ( n 2) P(E i ) P(E i1 E i2 ) + i=1 i 1 <i 2 +( 1) r+1 ( n r) i 1 <i 2 < i r P(E i1 E ir ) + + ( 1) n+1 P(E 1 E n ).

19 Sample Space Having Equally Likely Outcomes In many experiments, all possible outcomes in the sample space are equal likely to occur. Consider an experiment with finite sample space S = { 1,2,...,N }, then P ( {1} ) = P ( {1} ) = = P ( {N} ), which implies that P ( {i} ) = 1 N, i = 1,...,N. For any given event E in the sample space S Proof: P(E) = number of outcomes in E number of outcomes in S = n(e) N.

20 Example 1.18 If 3 balls are randomly drawn from a bowl containing 5 green (G) balls, 8 white (W) balls, and 7 black balls. Then, (1) what is the probability that 3 ball are all of a different color? (2) what is the probability that these is at least 1 green balls drawn? Solution: Example 1.19 What is the probability of obtaining four of a kind in a randomly dealt 5 card poker hand? Solution:

4. Probability of an event A for equally likely outcomes:

4. Probability of an event A for equally likely outcomes: University of California, Los Angeles Department of Statistics Statistics 110A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment: