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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} 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)} 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 4 n 6 } • 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 E S ) consisting of equally likely outcomes, the probability of E, which is denoted P(E) is |E| P( E ) |S| 9 The Probability • The following properties are important: 0 | E | M , 0 | S | N |E| 0 1 0 P( E ) 1 |S| 10 The Probability • The following properties are important: If P( E ) p P( E ) 1 p P( E E) 1 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 4 n 6 } 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 5 |S|= C52 C (52,5) 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 3 C selections of 3 pens chosen from 12: 12 C (12,3) • The favorable event E is to chose 3 pens among 10 nondefective ones C103 C (10,3) C103 C (10,3) 10! 12! 120 6 • P(E)=|E|/|S|= 3 / C12 C (12,3) 7!3! 9!3! 220 11 15 Homework • Read Section 8.5 paying a closer attention to examples. 16