Basic Concepts of Discrete Probability 1

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. 2

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 3

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. 4

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. 5

Sample Space We toss a single die, what are the possible outcomes, which form the sample space? {1,2,3,4,5,6} 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)} We toss a pair of dice, what is the sample space?

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} 7

The Probability The classical definition given by Laplace says that the probability is 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. 8

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 9

The Probability The following properties are important: 10

The Probability The following properties are important: 11

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 12

Coin experiment 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 13

Card experiment 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)

“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|= 15

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