Basic Concepts of Discrete Probability

1. Basic Concepts of Discrete Probability

2. Sample Space • When “probability” is applied to something, we usually mean an experiment with certain outcomes. • An outcome is any one of the possibilities that may be expected from the experiment. • The totality of all these outcomes forms a universal set which is called the sample space.

3. Sample Space • For example, if we checked occasionally the number of people in this classroom on Wednesday from 11am to 12-15pm, we should consider this an experiment having 19 possible outcomes {0,1,2,…,13,19} that form a universal set. • 0 – nobody is in the classroom, … 19 – all students taking the Discrete Mathematics Class and the instructor are in the classroom

4. Sample Space • A sample space containing at most a denumerable number of elements is called discrete. • A sample space containing a nondenumerable number of elements is called continuous.

5. Sample Space • A subset of a sample space containing any number of elements (outcomes) is called an event. • Null event is an empty subset. It represents an event that is impossible. • An event containing all sample points is an event that is certain to occur.

6. Sample Space We toss a single die, what are the possible outcomes, which form the sample space? {1,2,3,4,5,6} We toss a pair of dice, what is the sample space? Depends on what we’re going to ask. Often convenient to choose a sample space of equally likely events. {(1,1),(1,2),(1,3),…,(6,6)}

7. Sample Space • The following sets are subsets of the sampling set {1, 2, 3, 4, 5, 6} in the die-tossing experiments and therefore they are the events: • A={1, 2, 4, 6} • B={n: n is an integer and } • C={n: n is an even positive integer less than 7}

8. The Probability • The classical definition given by Laplace says that theprobabilityis the ratio of the number of favorable events to the total number of possible events. • All events in this definition are considered to be equally likely: e.g., throwing of a true die by an honest person under prescribed circumstances… • …but not checking the number of people in the classroom.

9. The Probability • According to the Laplace definition, for any event E in a finite sample space S (recall that if E is an event then ) consisting of equally likely outcomes, the probability of E, which is denoted P(E) is

10. The Probability • The following properties are important:

11. The Probability • The following properties are important:

12. Die-tossing experiments • Let us find the probabilities of the following events in the die-tossing experiments. • The sampling space is S={1, 2, 3, 4, 5, 6} • A={1, 2, 4, 6} P(A)=|A|/|S|=4/6=2/3 • B={n: n is an integer and } P(B)=|B|/|S|=2/6=1/3 • C={n: n is an even positive integer less than 7} P(C)= |C|/|S|=3/6=1/2

13. Coinexperiment • Let us flip a properly balanced coin three times. What is the probability of obtaining exactly two heads? • Each flip of the coin has two possible results (H) or (T) => according to the multiplication principle there are 2x2x2=8 possible outcomes for 3 flips S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} , three of which are favorable E={HHT, HTH, THH} => P(E)=|E|/|S|=3/8

14. Cardexperiment What is the probability that a 5 card poker hand contains a royal flush? S = all 5 card poker hands. A = all royal flushes P(A) = |A|/|S| |A|=4 |S|= P(A) = 4/C(52,5)

15. “Pen” experiment • Suppose that there are 2 defective pens in a box of 12 pens. If we choose 3 pens in random, what is the probability that we do not select a defective pen? • The sample space S consists of all possible selections of 3 pens chosen from 12: • The favorable event E is to chose 3 pens among 10 nondefective ones • P(E)=|E|/|S|=

16. Homework • Read Section 8.5 paying a closer attention to examples.