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Week 9 - Friday. CS322. Last time. What did we talk about last time? Partial orders Total orders Basic probability Event Sample space Monty Hall Multiplication rule. Questions?. Logical warmup.
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Week 9- Friday CS322
Last time • What did we talk about last time? • Partial orders • Total orders • Basic probability • Event • Sample space • Monty Hall • Multiplication rule
Logical warmup • There are 10 thieves who have just stolen an enormous pile of loot: gold, jewels, solid state drives, and so on • The thieves need to find a way to divide it all equally • Give an algorithm such that each of the 10 thieves believe that he is getting at least 1/10 of the loot • Hint: When you were a kid, how did your mother have you and your brother or sister divide the last piece of cake?
Permutations of letters in a word • How many different ways can the letters in the word "WOMBAT" be permuted? • How many different ways can "WOMBAT" be permuted such that "BA" remains together? • What is the probability that, given a random permutation of "WOMBAT", the "BA" is together? • How many different ways can the letters in "MISSISSIPPI" be permuted? • How many would it be if we don't distinguish between copies of letters?
Permuting around a circle • What if you want to seat 6 people around a circular table? • If you only care about who sits next to whom (rather than who is actually in Seat 1, Seat 2, etc.) how many circular permutations are there? • What about for n people?
Permutations of selected elements • An r-permutation of a set of n element is an ordered selection of r elements from the set • Example: A 2-permutation of {a, b, c} includes: • ab • ac • ba • bc • ca • cb • The number of r-permutations of a set of n elements is P(n,r) = n!/(n – r)!
r-permutation examples • What is P(5,2)? • How many 4-permutations are there in a set of 7 objects? • How many different ways can three of the letters in "BYTES" be written in a row?
Addition rule • If a finite set Aequals the union of k distinct mutually disjoint subsets A1, A2, … Ak, then: N(A) = N(A1) + N(A2) + … + N(Ak)
Addition rule example • How many passwords are there with length 3 or smaller? • Assume that a password is only made up of lower case letters • Passwords with length 3 or smaller fall into 3 disjoint sets • Number of passwords with length 1 • Number of passwords with length 2 • Number of passwords with length 3 • Total passwords = 26 + 262 + 263 = 18278
Difference rule • If A is a finite set and B is a subset of A, then N(A – B) = N(A) – N(B) • Example: • Recall that a PIN has 4 digits, each of which is one of the 26 letters or one of the 10 digits • How many PINs contain repeated symbols? • What is the probability that a PIN contains a repeated symbol?
Inclusion/exclusion rule • If A, B, C are any finite sets, then N(A B) = N(A) + N(B) – N(A B) • And, N(A B C) = N(A) + N(B) + N(C) – N(A B) – N(A C) – N(B C) + N(A B C)
Inclusion exclusion example • How many integers from 1 through 1,000 are multiples of 3 or multiples of 5? • How many integers from 1 through 1,000 are neither multiples of 3 nor multiples of 5?
Inclusion exclusion example • Consider a survey of 50 students about the programming languages they know • The results are: • 30 know Java • 18 know C++ • 26 know ML • 9 known both Java and C++ • 16 know both Java and ML • 8 know both C++ and ML • 47 know at least one of the three • How many students know none of the three? • How many students know all three? • How many students know Java and C++ but not ML? • How many students know Java but neither C++ nor ML?
Combinations Student Lecture
Subsets of sets • How many different subsets of size r can you take out of a set of n items? • Subset of size 3 out of a set of size 5? • Subset of size 4 out of a set of size 5? • Subset of size 5 out of a set of size 5? • Subset of size 1 out of a set of size 5? • This is called an r-combination, written
Permutations and combinations • In r-permutations, the order matters • In r-combinations, the order doesn't • Thus, the number of r-combinations is just the number of r-permutations divided by the possible orderings
Combinations example • How many ways are there to choose 5 people out of a group of 12? • What if two people don't get along? How many 5 person teams can you make from a group of 12 if those two people cannot both be on the team?
Poker examples • How many five-card poker hands contain two pairs? • If a five-card hand is dealt at random from an ordinary deck of cards, what is the probability that the hand contains two pairs?
r-combinations with repetitions • What if you want to take r things out of a set of n things, but you are allowed to have repetitions? • Think of it as putting r things in n categories • Example: n = 5, r = 4 • We could represent this as x||xx|x| • That's an rx's and n – 1 |'s
r-combinations with repetitions • So, we can think of taking an r-combination with repetitions as choosing r items in a string that is r + n – 1 long and marking those as x's • Consequently, the number of r-combinations with repetitions is
Example • Let's say you grab a handful of 10 Starbursts • Original Starbursts come in • Cherry • Lemon • Strawberry • Orange • How many different handfuls are possible? • How many possible handfuls will contain at least 3 cherry?
Handy dandy guide to counting • This is a quick reminder of all the different ways you can count things:
Next time… • Binomial theorem • Probability axioms • Expected values
Reminders • Work on Homework 7 • Due tonight before midnight • Keep reading Chapter 9
