This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Recap • Venn diagrams and deﬁnitions: union, intersection, subset, mutually exclusive • Properties of set operations • Deﬁnition of a probability function • Example: fair coin, dart board • Some theorems and inequalities Today’s outline • Leftover from last time: Boole’s inequality • Counting: combinations and permutations • Sampling with and without replacement • Enumerating outcomes and calculating probabilities • Examples: birthday problem, poker hands. Boole’s Inequality If P is a probability function • P (A) = • P (∪∞ ) ≤ i=1 Counting 1.2.12 How likely is it to win the lottery? 1.2.13 What is the probability of winning a knockout tournament with 16 participants? Permutations and combinations • In how many ways can 5 people form a team of 3? This is a combination. • What if they also have to decide who is President, vice President and helper? This is a permutation. Methods of counting • Ordered, without replacement • Ordered, with replacement • Unordered, without replacement • Unordered, with replacement Deﬁnitions • n! = n × (n − 1) × (n − 2) × . . . × 2 × 1 •
n r ∞ i=1 P (A ∩ Ci ) for any partition C1 , C2 , . . .. P (Ai ) for any sets A1 , A2 , . . .. ∞ i=1 = n! r !(n−r )! Without replacement Ordered Unordered Examples
n! (n−r )! n r With replacement nr
n+r −1 r • You are in a room of 30 people. What are the odds at least 2 of those have the same birthday? • You are playing a hand of poker (5 card stud). What are the odds you are dealt 4 aces? Homework logistics • At end of each lesson, you will be given problems to work on which will be due once a week. • You are free (and encouraged) to discuss the problems. However, the work you submit should be your own. • At the end of each lesson, you will be given some mathematical tools to refresh which we will use in the following lesson. • Tentative notes for next lecture will be uploaded on Blackboard before every lecture. 2 ...
View
Full
Document
This note was uploaded on 01/16/2011 for the course STAT 213 taught by Professor Ioannam during the Fall '09 term at Duke.
 Fall '09
 IoannaM
 Mutually Exclusive, Probability

Click to edit the document details