### Combinatorics & Probability

```Combinatorics & Probability
Section 3.4
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.
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, 4C3=4.
• How many ways to choose the 8s? ANSWER 4 choose 2,
4C2=6.
• Now multiply 6 and 4 and you get the number of ways of
getting Kings full of 8’s 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 13x4x12x6=3744.
• What is the probability of getting a full house?
Let’s 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= 52!
(52  5)!5!
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 can’t be kings so 48 choose 2.
• Thus we have 4C3=4 and 48C2=1128
• Therefore the probability of having a poker hand with
exactly 3 kings is 4 C3 48 C 2 =4512/2598960=0.001736
52
C5
Let’s 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 54912/2598960=0.0211=2.11%.
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
don’t 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 don’t re count the straight flushes and royal
flushes.)
• Compute the probability and odds of each.
```