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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

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

```
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