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CSE 103 > Lect 5: Combinatorics 3: Poker > Flashcards

Lect 5: Combinatorics 3: Poker Flashcards

1
Q

You pick one card from a shuffled deck. What is the probability that it’s rank is higher than 5?

A

Assuming that Ace is the highest we get 9/13

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2
Q

What are the ranks of poker hands?

A
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3
Q

What are the basic rules of texas hold’em poker?

A
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4
Q

How do betting rounds work in texas hold’em poker?

A
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5
Q

Calculating the probabilities of different hands. What is the sample space?

A

The sets of 5 cards out of 52. Order does not matter C(52,5) = 2,598,960

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6
Q

What is the probability of getting a royal flush?

A
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7
Q

What is the probability of a straight flush?

A
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8
Q

What is the probability of a four of a kind?

A
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9
Q

What is the probability of a full house?

A
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10
Q

What is the probability of getting a flush?

A
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11
Q

What is the probability of a straight?

A
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12
Q

What is the probability of three of a kind?

A
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13
Q

What is the probability of two pair?

A
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14
Q

What is the probability of one pair?

A
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15
Q

Poker hand table

A
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16
Q

What are the properties of general probability distributions?

A
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17
Q

How can you partition a union of two sets A ad B?

A
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18
Q
A
19
Q
A
20
Q

What is the general formula for

A
21
Q

What is the general formula for the probability of the union of 3 sets?

A
22
Q

What is the total probability equation for (countably) infinite sets?

A
23
Q

[WebWork 3.1.1] Find the number of distinguishable ways of arranging the letters “MAMA”

A

4! / (2!2!)

24
Q

[WebWork 3.1.2] Find the number of distinguishable permutations formed by using each of the letters AAABBC once and only once.

A

6! / (3!2!)

25
Q

[WebWork 3.2]

A
26
Q

[WebWork 3.3] Here we will calculate the probability of a student getting at least 60 percent on a true/false exam with 10 questions.

The probability that a student gets a grade of 60 percent or better by guessing is

A
27
Q

[WebWork 3.4] Suppose a seqeunce of 4 digits in the range 1-5 is chosen uniformly at random. What is the probability that the first and third digit are equal, the second and the fourth digits are equal, but the first and second digits are unequal?

A

|Ω| = 5^4

  • we have choices in each of the four locations the number of sequences, which is equal to the size of the sample space
  • we have 5 choices for the first digit then four choices for the second digit

P(A) = (5!/3!) / 5^4

A | = 5!/3!

28
Q

[WebWork 3.5] You are given two decks of numbered cards, each deck consists of 3 cards numbered from 3 to 5.

An experiment consists of drawing a card from each deck and summing their values. The sum is the outcome of the experiment. What is the outcome space?

A

The outcome space is {6,7,8,9,10}.

  • Recall that the outcome space is the set of all possible outcomes. Since one card draw can be any of {3,4,5}, the outcome space of the sum of two draws includes all possible results obtained by summing up two draws from {3,4,5}
  • {3+3,3+4,3+5,4+5,5+5}
29
Q

[WebWork 3.6] Suppose you have a deck with 10 different cards. How many ways are there to choose 3 cards out of this deck? (the order of the 3 picked out cardsdoes matter)

A

10! / (10-3)! = 10! / 7!

30
Q

[WebWork 3.7]

Consider a list of randomly generated 3-letter “words” printed on a paper. The letters cannot be repeated.

(a) What is the size of the set of allowed words?
(b) At least how many of these “words” should be printed to be sure of having at least 4 identical “words” on the list?
(c) At least how many identical “words” are printed if there are 78001 “words” on the list?

A

b) Recall the “Pigeon Hole” principle. If the number of different words is n then in order to ensure that at least one of the words appears at least k times you need to print (k−1)n+1 words. Now, if you can set k and n to the correct values, you’ll have the answer.
c) In this case you have to use the pigeon hole formula in the reverse direction. Find the value of kthat will be satisfied if you have 78001 words.

31
Q

[WebWork 3.8.1] How many sequences of length k (where order does matter) can we choose from a set of ndifferent elements?

A

n! / (n-k)!

  • sampling without replacement when the order matters
  • aka permutation function
32
Q

[WebWork 3.8.2] How many ways are there to order k elements?

A

k!

  • factorial function
33
Q

[WebWork 3.8.3] What is the binomial formula?

A
  • C(n,k)
  • n! / (k! (n-k)!)
  • aka choose
  • aka combinatoric function
  • aka binomial coefficient
34
Q

[WebWork 3.8.4] Suppose we have 6 bins, numbered 1,…,6 and that we have 6 balls, 3 of them white and 3 of them black.

How many white/black patterns can one make by placing the balls in the bins?

A

6! / (3! (6-3)!) = 6! / (3!3!) = 20

35
Q

How do you use stars and bars?

Use the following example:

You walk into a candy store and notice that there are five types of candy. Your mother allows you to pick exactly three pieces of candy, of whichever type(s) you want. How many ways are there to do this?

A
  • so bars equals n - 1
  • stars = the number you can pick, k
36
Q

[WebWork 3.9] Suppose balls come in 12 colors and that you are to pick out 2 balls. How many different color combinations are possible?

A

How many bars do you need?

  • n -1 = 11

How many stars do you need?

  • k = 2

What is the number of color combinations?

  • C(13,2)
37
Q

[WebWork 3.10.1]

Suppose we have 14 cards and 13 envelopes to put them in. The envelopes are marked (1,…,13).

Suppose all the cards are distinct (i.e. are numbered (1,2,….,14)). How many ways are there to place cards into envelopes?

A

13^14

  • Counting the number of combinations we get this way is simple. At each of the 14 step we place one card in one of the 13 envelopes. Taking the product over all of these steps we get that the number of combinations
38
Q

[WebWork 3.10.2] Suppose we have 14 cards and 13 envelopes to put them in. The envelopes are marked (1,…,13). Suppose that cards are identical. (The envelopes remain distinct) How many combinations are possible in this case

A

C(13−1+14,14)

  • Here the cards are identical, therefor we can only say how many cards are in each envelope, but we cannot identify them.

Thinking of the problem this way, we realize that it is mathematically equivalent to the problem of choosing 14 candies when there are 13 types of candy to choose from. As the Candies are indistinguishable, we are only interested in the number of candies chosen from each type. The correspondence is card <-> candy and candy type <-> envelope.

39
Q

[WebWork 3.10.3] Suppose we have 14 cards and 13 envelopes to put them in. The envelopes are marked (1,…,13). If all the cards are identical and each envelope contains at least one card, how many combinations are possible?

A
  1. take 13 cards and put one card in each envelope
  2. now we have a remaining one card w/ no constraints.
  3. We can put this 1 card in any of the 13 envelopes and follow the conditions

C(14-1, 14-13)

40
Q

[WebWork 3.11] A poker hand consisting of 4 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 3 cards of the same suite. It is allowed to have any number of cards in other suites.

A
41
Q

[WebWork 3.12] What is the probability of all poker hands?

A

C(52,5)

42
Q

[WebWork 3.13]

  1. What is the number of possibilities for the ranks of the two pair?
  2. What is the number of possibilities for the rank of the single?
  3. What is the number of possibilities for the suits of the two pair?
  4. What is the number of possibilities for the suit of the single?
  5. Thus the number of hands with exactly two pairs is?
  6. The probability of getting two hands is?
A
43
Q

[WebWork 3.14]

You have a deck that has 4 suits and 10 ranks.

Straight : Five cards in sequence, but not all from the same suit

  1. What is the number of possibilities for the ranks of a straight in this kind of deck?
  2. What is the number of possibilities of suits of a straight?
  3. What is the number of possible hands that will give o a straight?
  4. What is the probability of getting a straight in this deck?
A
44
Q

[WebWork 3.15]

In a regular deck, a full house is 2 of one rank and 3 of another rank.

  1. What is the number of possibilities for the rank of the triple?
  2. Given the rank of the triple, what is the number of possibilities for the rank of the pair?
  3. What is the number of possibilities for the suit of the triple?
  4. What is the number of possibilities of the suit of the pair?
  5. Thus the number of hands that is a full house is?
  6. What is the probability of getting a full house?
A

CSE 103 (10 decks)

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