- #1

CaptainK

- 14

- 0

## Homework Statement

You will assess the probability of a certain hand winning after the starting hands are dealt. You only need to consider two players at a time. (At no point are we considering betting or folding playing a part in winning)

Starting Points

-Try and watch some games of Texas hold 'em either on television or youtube. They show probabilities throughout each game

-Look at different possible starting hands, eg pairs, same suit...etc.

-Look at the following example with two players;

If player 1 has two fives (a heart and a spade) as his starting hand and player 2 has an ace of spades and a king of hearts.

-What is the probability that player 1 will win this deal after the 5 community cars are dealt?

-Select some different starting hands, two at a time, and calculate the probability that one hand will win - this will form the majority of your assignment.

## Homework Equations

^{n!}/

_{(n-r)!}for permutations

^{n!}/

_{(n-r)!r!}for combinations

## The Attempt at a Solution

So far I have done mere scratchwork on the problem as I am not 100% sure where to start or what i should do.

I have selected two random hands which are as follows

hand 1 consists of a 7 of spades and a 4 of hearts

hand 2 consists of a 5 of diamonds and a Jack of diamonds

this hand is good because there are no pairs, no two cards follow each other numerically and it incorporates two crads of the same suit.

I know that combinations are needed to complete this problem as order does not matter when the community cards are turned up.

The probability of each hand winning does not add up to 100% like it should, because of the small chance that the hands tie.

I also know that i need to go through the chances of each of the hands winning with the different combinations as shown below...

High Card (no special combinations)

One Pair (eg two jacks

Two Pair (eg two aces and two fours)

Three of a Kind (eg three queens)

Straight ( all 5 cards in order, eg 7, 8 ,9 ,10, J)

Flush (all 5 cards the same suit)

Full House (three of a kind and a pair)

Four of a kind (eg four kings)

Straight Flush (all 5 cards in order and of the same suit)

I know I need to calculate this doing combinations, but I also know that there are over 1 million different combinations of cards that could turn up and instead of calculating each of them out, if there is a shorter way to do it?

If not are there any shortcuts I can take?

Any help is greatly appreciated as I missed the lessons on probability and can use any help that I can get.

Cheers