Chp 4 Notes: Finite Uniform Probability Spaces Flashcards
What are the components of finite uniform probability spaces?
What are events?
Events are subsets of Ω
What does it mean to have finite unfiorm distribution? How do we find probability of finite sets that have uniform distribution?
What is the probability of getting a flush in a five-card poker?
Events are subsets of Ω, in our example, the event of interest is A = {ω : ω is a flush}. This is a subset of Ω; that is, A ⊂ Ω.
Roll a die. What is the chance of getting a number > 3?
Roll three dice. What is the chance their sum is 3?
Roll n dice. What is the sample space? What is the probability of any one outcome?
Socks in a drawer. A drawer contains three blue socks and three red socks. You put your hand in and pick out a random sock. Then you put your hand in again and pick out another random sock. What’s the chance the two of them match?
Socks in a drawer, again. This time the drawer has three blue socks and four red socks.You put your hand in and pick out a random sock. Then you put your hand in again and pick out another random sock. What’s the chance the two of them match
Shuffling a deck of cards. You randomly shuffle a deck of 52 cards and lay them out before you. What is the outcome space?
Toss a fair coin 10 times. What is the sample space? What is the probability of any outcome?
Toss a coin 10 times. What is the chance that none of the coin tosses are heads?
The event of interest is {(T,T,T,T,T,T,T,T,T,T)}, whose probability is 1/1024.
Toss a coin 10 times. What is the chance of exactly one head?
Toss a coin 10 times. What is the chance of exactly nine heads?
Equivalently, what is the chance of exactly one tail? This is the same calculation as before, 10/1024.
Toss a coin 10 times. What is the chance of exactly two heads?
Toss a coin 10 times. What is the chance of exactly four heads?
Toss a coin 10 times. What is the chance of exactly six heads?
What is the binomial coefficient?
The combinatorial number, C( n, k), the number of ways of picking unordered outcomes from n possibilities.
What does disjoint mean?
Notice that the events A0, A1, . . . , An are disjoint (if Ai occurs then Aj cannot occur for j ̸= i)
Ex. if event of a coin toss was {T} then it could not have been {H}
Toss a fair coin n times. Now the sample space is Ω = {H,T}n, with each sequence of n outcomes having probability exactly 1/2n. Let Ak denote the event that the sequence has k heads. What is the size of Ak?
Rooks on a chessboard. You place 8 rooks at random on a chessboard. What is the chance that they are non-attacking (that is, no rook is attacking another)?
Birthday paradox. A room contains n people. What is the chance that two of them have the same birthday?
Assume probability of birthday is 1/365 and each person’s birthday is independent of each other.
Balls in bins. You have m indistinguishable balls and in front of you is a row of n bins. You place each ball into a bin chosen at random.
Let’s write the sample space as Ω = {1,2,…,m}n; in each outcome ω = (ω1,…,ωn), the value ωi represents the number of balls in the ith bin.
Here are some interesting tidbits to prove.
The chance that any particular bin is empty is at most?
Balls in bins. You have m indistinguishable balls and in front of you is a row of n bins. You place each ball into a bin chosen at random. If m = 2n ln n, the chance that there exists an empty bin is at most?
Balls in bins. You have m indistinguishable balls and in front of you is a row of n bins. You place each ball into a bin chosen at random. The chance that no bin has 2 (or more) balls is at most?
What is this problem an example of?
For instance, tossing a fair coin m times
- it is a simple case of balls and bin
- tossing a fair coin m times is like throwing m balls into n = 2 bins (call one bin H and the other bin T ).
You are dealt five cards at random from a deck of 52 cards. What is the chance of a flush?
Let Ω = {all possible 5-card hands}. Then |Ω| = C(52, 5) and each ω ∈ Ω occurs with probability 1/|Ω|.
F = flush (all five cards of the same suit)
You are dealt five cards at random from a deck of 52 cards. What are the chances of a straight flush?
Let Ω = {all possible 5-card hands}. Then |Ω| = C(52,5) and each ω ∈ Ω occurs with probability 1/|Ω|.
S = straight flush (same suit and consecutive)
(first choose a suit, then choose a starting card in the sequence)
S | = 4 * 10
You are dealt five cards at random from a deck of 52 cards. What is the chance of exactly one pair?
Let Ω = {all possible 5-card hands}. Then |Ω| = C(52,5) and each ω ∈ Ω occurs with probability 1/|Ω|.
P = the cards contain a single pair (eg. two 7s)
|P| = 13 * C(4, 2) * 43 * C(12,3)
- first choose which card occurs in the pair
- then choose the two suits for that pair
- then choose the suits of the remaining three cards
- then choose their values
