Chp 4 Notes: Finite Uniform Probability Spaces

Question 1

What are the components of finite uniform probability spaces?

Answer 1

Question 2

What are events?

Answer 2

Events are subsets of Ω

Question 3

What does it mean to have finite unfiorm distribution? How do we find probability of finite sets that have uniform distribution?

Answer 3

Question 4

What is the probability of getting a flush in a five-card poker?

Answer 4

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.

Question 5

Roll a die. What is the chance of getting a number > 3?

Answer 5

Question 6

Roll three dice. What is the chance their sum is 3?

Answer 6

Question 7

Roll n dice. What is the sample space? What is the probability of any one outcome?

Answer 7

Question 8

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?

Answer 8

Question 9

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

Answer 9

Question 10

Shuffling a deck of cards. You randomly shuffle a deck of 52 cards and lay them out before you. What is the outcome space?

Answer 10

Question 11

Toss a fair coin 10 times. What is the sample space? What is the probability of any outcome?

Answer 11

Question 12

Toss a coin 10 times. What is the chance that none of the coin tosses are heads?

Answer 12

The event of interest is {(T,T,T,T,T,T,T,T,T,T)}, whose probability is 1/1024.

Question 13

Toss a coin 10 times. What is the chance of exactly one head?

Answer 13

Question 14

Toss a coin 10 times. What is the chance of exactly nine heads?

Answer 14

Equivalently, what is the chance of exactly one tail? This is the same calculation as before, 10/1024.

Question 15

Toss a coin 10 times. What is the chance of exactly two heads?

Answer 15

Question 16

Toss a coin 10 times. What is the chance of exactly four heads?

Answer 16

Question 17

Toss a coin 10 times. What is the chance of exactly six heads?

Answer 17

Question 18

What is the binomial coefficient?

Answer 18

The combinatorial number, C( n, k), the number of ways of picking unordered outcomes from n possibilities.

Question 19

What does disjoint mean?

Answer 19

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}

Question 20

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 A_k denote the event that the sequence has k heads. What is the size of A_k?

Answer 20

Question 21

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)?

Answer 21

Question 22

Birthday paradox. A room contains n people. What is the chance that two of them have the same birthday?

Answer 22

Assume probability of birthday is 1/365 and each person’s birthday is independent of each other.

Question 23

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?

Answer 23

Question 24

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?

Answer 24

Question 25

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?

Answer 25

Question 26

What is this problem an example of?

For instance, tossing a fair coin m times

Answer 26

• 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 ).

Question 27

You are dealt five cards at random from a deck of 52 cards. What is the chance of a flush?

Answer 27

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)

Question 28

You are dealt five cards at random from a deck of 52 cards. What are the chances of a straight flush?

Answer 28

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

Question 29

You are dealt five cards at random from a deck of 52 cards. What is the chance of exactly one pair?

Answer 29

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)

1. first choose which card occurs in the pair
2. then choose the two suits for that pair
3. then choose the suits of the remaining three cards
4. then choose their values