Question

Consider a Poker game where an opponent tells you that the five cards she or he holds (and which you cannot see) represent a hand that beats 60% of all other possible hands. (If you aren't that familiar with Poker, a quick web search will help you understand the relationships between terrible and excellent poker hands -- with better hands having lower probability) Discuss how you would use that knowledge, along with the Hypergeometric Distribution, to correctly identify the hand your opponent is holding. Be very specific in describing your step by step approach to solving the problem. Exactly what probabilities would you be calculating, in what order, and why? How accurate can you be, and why can’t you be more accurate than that?

Answer 1

Generally in a poker game there are 10 hands with each hand consisting of 5 cards which are ranked appropriately.

The ranking of the hands are in this order

1st , 2nd , 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th

where 1st is the highest ranked hand and win over 2nd - 10th hand

and so is 2nd higher that 3rd and so on

Now 60% of 10 hands is 6 hands,this means that your opponent can win 6 hands of all the 10 possible hands hence the five card or hand that he or she is holding is ranked 4th among the 10 hands.This would mean that he/she would win any hand that is ranked from 5th to 10th.

Now for you to be able to win him/her you must be holding a hand that is of higher ranking than 4th and these are 1st ,2nd,3rd.When we play any of this hand having this ranks you would be able to win or draw the game.

Now if you choose the fourth ranked hand cause it easier to get than the 1st second and third hand ranked cards,this hand which is called "full house" is made up of one pair and "a three of a kind"which is of different rank than one pair but when combined it makes up a hand that is ranked 4th,when you play this card and if what he/she said its true then you will draw the game.In the case that you desire to win the game then you need to have a play a hand that is ranked 3, 2 ,1 for a hand ranked 3 which is called a "Four-of-kind" and in this hand all the foul cards are of the same rank, this hand when played in this situation would give you a win.We can see that following this technique we can win or draw this game .In the case that your opponent lied then we can say that at least he/she can win 60% of the possible hand which mean that she might be holding a hand ranked 4th called Full house or a hand ranked 3rd call four of a kind or a hand ranked 2nd called Straight Flush or a hand ranked 1st called Royal flush this would be in the case he/she lied.Looking at these techniques we can see that we can not be more accurate than this as this are the only possibilities.

Explanation:

In order to get a better understanding of the above answer a note where Hypergeometric Distribution was used explain what Hypergeometric Distribution is

Hypergeometric Distribution:

This can be defined in statistics as a discrete probability distribution that explains the probability for a specific number z of success to occur.in n draws, without replacement, from a finite population of size N that contains exactly Z objects with that feature, wherein each draw is either a success or a failure.Note that each object that is been draw has a specific feature.

Now looking at our technique which we applied to determine how to win our opponent or how our opponent would win the game in a situation where he or she lied we can see that we applied Hypergeometric Distribution knowledge which helped us to understand that each hand played has a feature specific to it and that is their ranking