Lecture 6: Probability. If S is a sample space with all outcomes equally likely, define the probability of event E,


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1 Lecture 6: Probability Example Sample Space set of all possible outcomes of a random process Flipping 2 coins Event a subset of the sample space Getting exactly 1 tail Enumerate Sets If S is a sample space with all outcomes equally likely, define the probability of event E, P(E) = the number of outcomes in E = n(e) the total number of outcomes in S n(s) ex. in flipping 2 coins, what s the probability of getting exactly 1 tail? ex. in flipping a coin what s the probability of getting a head? (What's E?) ex. in drawing a card, what s the probability of drawing a face card? Computing the Number of Possibilities When Order Counts ex. How many ways different 4letter strings can be formed with the alphabet a,b,c? ex. How many 3letter words can be formed? ex. How many ways can the Orioles and the Yankees win or lose a 3 game series? Lecture 13 p. 1
2 Multiplication Rule: If an operation consists of k steps step i k can be performed... in ways the entire operation can be performed in ways ex: How many user IDs are possible if each must consist of 5 characters and the first character must be an uppercase letter and the other 4 characters must be digits or uppercase letters? Some problems are not so simple: ex: How many results are there for the World Series? Game When Order Matters Using Trees to Determine the Number of Possible Outcomes (of a sequence of events) "Draw the tree and count the leaves" Example 1. How many sequences of 22 flips of a coin exist? Start Lecture 13 p. 2
3 Example 2: How many ways can you schedule your courses if CS can occur at 11 or 12, English at 10 or 2, and Physics at 10 or 1? Start [See the club officer example, pp ] Example 3: How many ways can a match in women s tennis proceed to its end? Lecture 13 p. 3
4 Permutations If all n distinct objects are rearranged in a sequence, this is called a permutation of the n objects. The number of permutations of n distinct elements is ex. How many ways can the 15 kindergarten children form a line? If only some, say r, of the n distinct objects are arranged in an ordered sequence, this is called an rpermutation. The number of rpermutations of a set of n distinct elements is ex. How many ways can 5 of the 15 kindergarten children be selected to form a line? Lecture 13 p. 4
5 Counting the Number of Elements in a Set (Remember, the elements in a set are NOT ordered.) ex. How many words are possible of length 4 or less? (assume only lowercase letters are used) Number of words of length 4 = Number of words of length 3 = ex. My grade report consists of 5 letter grades, one for each course, each an A, B, C, D, or F. How many possible grade reports will contain at least 1 A? Number of possible reports = Number of reports with at least 1 A= Suppose all the grade reports are equally likely. What s the probability that I ll get all A s? What is the probability of no A s? ex. I roll a die 3 times. How many possible sequences of rolls will contain at least 1 one? Number of possible sequences = Number of sequences with at least 1 one = What s the probability that I will roll at least 1 one? Lecture 13 p. 5
6 ex. In recent survey of 50 Loyola CS majors who can program in Java and/or C, 28 students said they can program only in Java and 12 said they can program only in C. How many can program in both? Inclusion/Exclusion Rule for Sets n(a B) = n(a) + n(b) n(a B) n(a B C ) = n(a) + n(b) + n(c) n(a B) n(a C) n(b C) + n (A B C) ex. In recent survey of 70 Loyola CS majors who can program in Java and/or C and/or Prolog, 18 students said they can program in Java and C and 10 said they can program in C and Prolog, 8 said they can program in Java and Prolog. 46 claimed to be able to program in Java, 38 in C, and 16 in Prolog. How many can program in all 3 languages? Lecture 13 p. 6
7 Combinations ex: Choose 2 representatives from our class for as CS council. How many different combinations are there? Does the order of the elements matter? In these examples, underline the "key" word that indicates that order does NOT matter. ex: How many pairs of socks are possible when selected from a drawer of 8 socks, each of a different color? ex: How many groups of 3 letters can be selected from the set { a, b, c, d }? ex: How many bridge foursomes are possible with members selected from this class? An rcombination of a set of n elements is a subset of r of the n elements. Denoted & n# $! % r " Lecture 13 p. 7
8 RELATIONSHIP BETWEEN PERMUTATIONS AND COMBINATIONS: Number of kcombinations = Number of kpermutations. The number of kpermutations containing the same set of elements & n # $! = % k " Read as " ex: How many groups of 6 students can I select from a group of 15 to ride in this car? ex: How many a groups of 4 students can be formed from a class of 3 females and 5 males in which exactly one of the students is female are possible? ex: Software Engineering teams of 4 members are being formed in a class of twenty students, of whom 6 are female and 14 are male. If all combinations are equally likely, what is the probability that a team will contain no females? Will contain exactly 1 female? Will contain at least 1 female? Will contain at most 1 female? Lecture 13 p. 8
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