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Chapter 2: Q. 2.4 (page 52)

Calculate the number of possible five-card poker hands, dealt from a deck of 52 cards. (The order of cards in a hand does not matter.) A royal flush consists of the five highest-ranking cards (ace, king, queen, jack, 10) of any one of the four suits. What is the probability of being dealt a royal flush (on the first deal)?

Short Answer

Expert verified

The chances of getting a royal flush on the first deal are slim Pflushes=1.539×10−6

Step by step solution

01

Step1:definition of probability

Probability is a measure of the likelihood of an event occurring. Many events are impossible to predict with 100% accuracy. We can only predict the likelihood of an event occurring using it, that is, how likely it is to occur. Probability can range between zero and one, with zero indicating an impossible event and one indicating a certain event.

02

Step2:Probability of royal flushes

Because there are52 cards in the deck, the number of ways to choose five cards from this deck is:

Ω(52,5)=52!5!(52−5)!=2598960

There are four royal flushes among these ways, so the probability of these flushes is:

Pflushes=42598960=1.539×10−6

Pflushes=1.539×10−6

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Most popular questions from this chapter

Consider a system of two Einstein solids, with N{A} = 300, N{B} = 200 and q{total} = 100 (as discussed in Section 2.3). Compute the entropy of the most likely macrostate and of the least likely macrostate. Also compute the entropy over long time scales, assuming that all microstates are accessible. (Neglect the factor of Boltzmann's constant in the definition of entropy; for systems this small it is best to think of entropy as a pure number.) 65

Use a computer to produce a table and graph, like those in this section, for the case where one Einstein solid contains 200 oscillators, the other contains100 oscillators, and there are 100 units of energy in total. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?

Using the same method as in the text, calculate the entropy of mixing for a system of two monatomic ideal gases, Aand B, whose relative proportion is arbitrary. Let Nbe the total number of molecules and letx be the fraction of these that are of speciesB . You should find

ΔSmixing=−Nk[xln⁡x+(1−x)ln⁡(1−x)]

Check that this expression reduces to the one given in the text whenx=1/2 .

The mixing entropy formula derived in the previous problem actually applies to any ideal gas, and to some dense gases, liquids, and solids as well. For the denser systems, we have to assume that the two types of molecules are the same size and that molecules of different types interact with each other in the same way as molecules of the same type (same forces, etc.). Such a system is called an ideal mixture. Explain why, for an ideal mixture, the mixing entropy is given by

ΔSmixing=kln⁡NNA

where Nis the total number of molecules and NAis the number of molecules of type A. Use Stirling's approximation to show that this expression is the same as the result of the previous problem when both Nand NAare large.

How many possible arrangements are there for a deck of 52playing cards? (For simplicity, consider only the order of the cards, not whether they are turned upside-down, etc.) Suppose you start w e in the process? Express your answer both as a pure number (neglecting the factor of k) and in SI units. Is this entropy significant compared to the entropy associated with arranging thermal energy among the molecules in the cards?
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