A hand of five-card draw poker consists of an unordered...

A hand of five-card draw poker consists of an unordered arrangement of five cards from an ordinary deck of 52 playing cards. a. How many five-card draw poker hands are possible? b. How many different hands consisting of three kings and two queens are possible? c. The hand in part (b) is an example of a full house: three cards of one denomination and two of another. How many different full houses are possible? d. Calculate the probability of being dealt a full house. e. Compare your answers in parts (a)-(d) to those in Exercise 4.187

A hand of five-card draw poker consists of an unordered arrangement of five cards from an ordinary deck of 52 playing cards.
a. How many five-card draw poker hands are possible?
b. How many different hands consisting of three kings and two queens are possible?
c. The hand in part (b) is an example of a full house: three cards of one denomination and two of another. How many different full houses are possible?
d. Calculate the probability of being dealt a full house.
e. Compare your answers in parts (a)-(d) to those in Exercise 4.187

Key Concepts

Combinatorial Counting

This concept involves determining the number of ways to select items from a larger set where the order of selection does not matter. In combinatorial problems, counting techniques such as combinations are used to calculate the total number of possible groups, which is fundamental when analyzing the number of card hands or any scenario where subsets are chosen.

Binomial Coefficients

Central to combinatorial counting, binomial coefficients represent the number of ways to choose a subset of items from a larger set and are denoted as 'n choose k'. This concept is crucial for determining the number of possible hands or arrangements when the order is irrelevant, and it serves as the mathematical underpinning for the formula used in many probability and counting problems.

Probability Calculation Using Combinatorics

This key concept deals with computing the probability of a specific event by determining the ratio of favorable outcomes to the total number of possible outcomes. By using combinatorial methods to count both the numerator (favorable cases) and the denominator (all possible cases), one can accurately calculate probabilities in situations such as card games or other selection-based scenarios.

Enumeration with Restrictions

This concept pertains to counting the number of arrangements or selections that meet specific criteria or restrictions, such as having a specified number of items of certain types. It involves breaking down the problem into distinct cases, applying combinatorial methods for each case, and then combining the counts. This technique is widely applicable in problems where subsets must fulfill predetermined conditions.

Transcript

00:01 In this problem, we are dealing with card drawing in a hand of five -card draw poker, which is an unordered arrangement of five cards from an ordinary depth of 52 playing cards, and we are being asked to calculate various probabilities regarding a five -card draw.

00:17 So in part a, we are being asked how many five -card draw poker hands are possible.

00:23 And as already mentioned, a five -card draw is a game where order is not important, so we need to use a combination.

00:33 And since there are 52 cards and we want five cards, we will do 52 choose 5, which is 52 factorial over 5 factorial times 52 minus 5 factorial.

00:46 Which is 52 factorial over 5 factorial 47 factorial.

00:51 And this is equal to 2 ,59 ,960 when you calculated out.

00:59 In part b, we are being asked that we are being asked how many different hands consisting of three kings and two queens are possible.

01:14 And what we want to do is select, since there are four kings and four queens, we want to select three of the kings and two of the queens.

01:23 And order, once again, is not important, so we use the combination.

01:26 So we do four choose three to select three kings out of four queens and multiply that by four choose two, which is selecting two queens out of four queens.

01:38 And this is four factorial over three factorial or minus three factorial times four factorial over two factorial or minus two factorial.

01:50 And if you calculate that out, we get four times six, which is equal to 24.

01:59 In part c, we are told that the hand in part b is an example of a full house...

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