26. Statistical Mechanics of Poker: in statistical mechanics, microstates associ

Question

26. Statistical Mechanics of Poker: in statistical mechanics, microstates associated with classical or quantum mechanics are effectively partitioned into macrostates. The definitions and specifications of these macrostates are not given to us by nature but are to some extent a matter of human invention although obviously some choices will be more useful than others, being more naturally associated with typical sorts of bulk measurements, or leading to more reproducible predictions But the situation is somewhat analogous to a card game like poker, where the microstates correspond to the particular cards dealt to given players and in a certain order, while the macrostates correspond to the classification of the hands, such as one pair, two pair, three of a kind, straight, flush, etc. These macrostates are human inventions, but obviously some choices of macrostates will be easier to remember than others, or will typically lead to more interesting games, etc. Consider a game of five-card stud, where each player receives five cards (with no opportunity for sub- stitutions) from a standard deck of 52 cards. The Boltzmann entropy is associated with the number of microstates compatible with a given macrostate, which is then here proportional to the probability for getting dealt a specified type of hand. These probabilities then also determines the rank of the possible hands in the game. (a) What is the probability of getting dealt a full house, exactly 3 cards with the same face value and the remaining 2 cards both sharing a common face value? (b) What is the probability of getting dealt a flush, with all five cards having the same suit? Of course, you can easily find these odds listed in books or online, but it is more fun and informative to actuallv work them out vourself But if history had played out differently, we could have introduced other types of poker hands (c) Calculate the probability of being dealt a skip-straight, in which the five cards constitute a sequence of successive even values or odd values for example 2,4,6,8,10. Face cards are ordered as usual (where Jack 11, Queen 12, King-13, and aces can be taken to have either a value of 1 or 14) (f) Explicitly define your own new type of five-card poker hand, and calculate its probability in a game of five-card stud. Where would this type of hand fall in the rankings of standard poker hands (which now you can look up)!

Explanation / Answer

Here you have to find the probability of flush in poker hand.

You have to choose 5 cards from 52 cards.

possibilities for 5 cards is

52C5 =2598960

a)

For full house 3 cards are same cards( eg. 3,3,3) and others two cards also same cards(eg .2,2)

We choose 1 card from 13 cards( 3)

13C1=13

3,3,3 are different suit so 4C3=4

Other card 2 is choose from remaining 12 crads so 12C1=12

And suit for this other cards is 4C2=6

So probability of full house = 13*4*12*6 / 2598960

=3744/2598960

=0.001441

b)

The total number of cards in sequence 1 to K is 13. you choose 5 cards ( eg 4,7,9,10,J)

13C5=1287

For flush choose each cards from same suit . total number of suit is 4 you choose 1 suit

4C1=4

probability of flush =1287*4 / 2598960

=5148/2598960

=0.001981

c)

In straight and royal flush are subtract from above .

10C1 * 4C1=40

probability flush skip straight = (5158 - 40 )/2598960

=0.001965

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