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 knock-out 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