Combinatorics

1
Combinatorics Probability

• Section 3.4

2
Which Counting Technique?

• If the problem involves more than one category,
  use the Fundamental Principle of Counting.
• Within any one category, if the order of
  selection is important use Permutations.
• Within any one category, if the order of
  selection is not important, use Combinations.

3
A Full House

• What is a full house? An example would be three
  Ks and two 8s. We would call this Kings full
  eights.
• How many full houses are there when playing 5
  card poker?
• First think of the example How many ways to
  choose 3 kings? ANSWER 4 choose 3, 4C34.
• How many ways to choose the 8s? ANSWER 4 choose
  2, 4C26.
• Now multiply 6 and 4 and you get the number of
  ways of getting Kings full of 8s which is 24.
• A full house is any three of kind with a pair. So
  take 24 and multiply by 13 (13 ranks for the
  three of a kind) and by 12 (12 ranks for the
  pair, note you used one rank to make the three of
  a kind).
• So the number of full house hands is
  13x4x12x63744.
• What is the probability of getting a full house?
• ANSWER 3744/25989600.001440.144

4
Lets Go Further and talk about a three of a kind

• What is the probability of having exactly three
  Kings in a 5-card poker hand.
• First, how many 5-card poker hands are there?
• ANSWER 52 choose 5 or 52C5
• which is 2,598,960
• Now how do we figure out a hand that has exactly
  3 kings?
• ANSWER There are 4 kings so we choose 3. The
  other 2 cards cant be kings so 48 choose 2.
• Thus we have 4C34 and 48C21128
• Therefore the probability of having a poker hand
  with
• exactly 3 kings is
  4512/25989600.001736

5
Lets go further

• What is the probability of being dealt a three of
  a kind. This is a little different from the last
  problem. Last problem we had a specific three of
  a kind, so now we can multiply the result by 13
  (since there are 13 ranks). So the number of
  hands that have a three of a kind in them is
  58656. Some of these hands are actually full
  houses. So we should subtract from this result.
  Which would give 54912. Hence the probability of
  being dealt a three of a kind (not a full house)
  is 54912/25989600.02112.11.

6
FLUSH

• Figure out the number of ways you can get a Royal
  Straight Flush (A,K,Q,J,10 of the same suit) in 5
  card poker.
• Figure out how many straight flushes you can get
  in 5 card poker. (example 8,7,6,5,4 of the same
  suit and dont recount the royal flushes.)
• NOTE THIS IS NOT A STRAIGHT Q,K,A,2,3 NO
  WRAPAROUND.
• Figure out how many flushes in a 5 card poker
  hand. (Note dont re count the straight flushes
  and royal flushes.)
• Compute the probability and odds of each.