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							Math 21, Tu1/06/09
                               Lecture / Homework 1
                            Introduction to Sets & Probability
I. The basics of probability
                                                 Number of successful outcomes in the event
    A. P ( A) = “The Probability of Event A” =
                                                  Total # of outcomes in the "sample space"
    B. Types of probability
       1. Theoretical (“Classical”): Either all outcomes are equally likely, or all outcomes can
          be broken down into parts that are equally likely. You can predict probabilities by
          enumerating all possible outcomes. Typical examples involve dice, coins, and
          playing cards.
       2. Empirical Probability (“Relative Frequency”): The results of a survey or experiment,
          usually summarized in a table. These values could not be predicted by theory.
       3. The Law of Large Numbers: When a predictable experiment is repeated more and
          more often, Empirical Probability gets closer and closer to Actual (Theoretical)
          Probability.

II. Relation to sets and counting principles
    A. The possible outcomes are usually grouped together in “sets,” and the number of
       outcomes is described as the size of the set. Counting principles are used to calculate the
       sizes of sets when they are too difficult to write out.
    B. We can and will study probability without using specialized set notation and language,
       but for reference this language is described in § 6.1.
    C. Counting principles will come in future lectures, after we have covered the basics of
       probability.

III. Basic AND & OR problems
     A. AND means just look at the intersection of two categories, e.g. a column and row
     B. OR means to add outcomes together from various events, e.g. find the total of a row or
        column.
     C. For now, we are assuming that the rows or columns are “mutually exclusive,” meaning
        they do not have any overlap.

IV. References for gambling-type questions
    A. Deck of cards: Refer to end of § 6.3
    B. Rolling a pair of dice: See § 7.2, Example 3, or lecture notes below.
    C. Flipping multiple coins: Use tree diagram (see § 7.2 Example 5) or see notes below.
Math 21, Lecture 1, 1/06/09



Reading
Section 6.1. Introduction to Sets: Optional background reference only.
Section 7.1. Introduction to Probability.
Section 7.2. Equally Likely Events (Theoretical Probability): Skip examples 4, 6, and 7.

Exercises
§ 7.1: #’s 11, 13, 19, 21, 25
§ 7.2: #’s 1, 3, 5, 11, 13


Reference tables


Flipping multiple coins

                                2 coins   3 coins    4 coins
                                HH        HHH        HHHH
                                HT        HHT        HHHT
                                TH        HTH        HHTH
                                TT        HTT        HHTT
                                          THH        HTHH
                                (4        THT        HTHT
                                outcomes) TTH        HTTH
                                          TTT        HTTT
                                                     THHH
                                           (8        THHT
                                           outcomes) THTH
                                                     THTT
                                                     TTHH
                                                     TTHT
                                                     TTTH
                                                     TTTT

                                                       (16
                                                       outcomes)
Math 21, Lecture 1, 1/06/09


Rolling a pair of dice


    There are 36 possible rolls arising in 12 possible sums. This table shows how many rolls are
    possible for each sum. See § 7.2 for more detail.

                                 Sum of 2 dice Ways To Roll
                                      2             1
                                      3             2
                                      4             3
                                      5             4
                                      6             5
                                      7             6
                                      8             5
                                      9             4
                                     10             3
                                     11             2
                                     12             1
                                   TOTAL           36