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Chapter 15: Q52P (page 412)

(III) Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostates.

Short Answer

Expert verified

The ranking in increasing order of probability is (b), (a), (c), and (d).

Step by step solution

01

Macrostate and microstate in a deck of cards

In this problem, the macrostate is a set of five cards from the deck.

Also, a microstate is a specific set of cards that meet the criterion of a given macrostate.

02

Estimating the order of four aces and a king

In this condition, there are four microstates for this macrostate, corresponding to the deck from which the king originates.

03

Estimating the order of six of hearts, eight of diamonds, queen of clubs, three of hearts, and jack of spades

In this condition, every individual card is specified. Hence, there is only 1 microstate for the macrostate in part (b).

04

Estimating the order of two jacks, two queens, and an ace

In this condition, there are six possible pairs of jacks and queens. For the ace, only four pairs are possible.

The total number of card combinations is calculated below:

Hence, there are 144 microstates for this macrostate.

05

Estimating the order of any hand having no two equal value cards

There are 52 possibilities for the first case, 48 for the next one, and so on. In this condition, it is apparent that there are many more microstates than for any of the other given macrostates. Hence, it has the highest probability.

Thus, the increasing order of probability is (b), (a), (c) and (d).

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Most popular questions from this chapter

Question: (II) A 65-kg person decides to lose weight by sleeping one hour less per day, using the time for light activity. How much weight (or mass) can this person expect to lose in 1 year, assuming no change in food intake? Assume that 1 kg of fat stores about 40,000 kJ of energy.

Refrigeration units can be rated in “tons.” A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton (2000 pounds = 909 kg) of 0°C water into 0°C ice in one 24-h day. If, on a 35°C day, the interior of a house is maintained at 22°C by the continuous operation of a 5-ton air conditioning system, how much does this cooling cost the homeowner per hour? Assume the work done by the refrigeration unit is powered by electricity that costs $0.10 per kWh and that the unit’s coefficient of performance is 18% that of an ideal refrigerator.\({\bf{1}}\;{\bf{kWh = 3}}{\bf{.60 \times 1}}{{\bf{0}}{\bf{6}}}\;{\bf{J}}\).

Question: (II) A heat pump is used to keep a house warm at 22°C. How much work is required of the pump to deliver 3100 J of heat into the house if the outdoor temperature is (a) 0°C, (b) \({\bf{ - 15^\circ C}}\)? Assume a COP of 3.0. (c) Redo for both temperatures, assuming an ideal (Carnot) coefficient of performance \({\bf{COP = }}{{\bf{T}}_{\bf{L}}}{\bf{/}}\left( {{{\bf{T}}_{\bf{H}}}{\bf{ - }}{{\bf{T}}_{\bf{L}}}} \right)\).

A “Carnot” refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of -17°C and exhausts it into the room at 25°C.

(a) How much work would the refrigerator do to change 0.65 kg of water at 25°C into ice at -17°C.

(b) If the compressor output is 105 W and runs 25% of the time, how long will this take?

Question: Which do you think has the greater entropy, 1 kg of solid iron or 1 kg of liquid iron? Why?
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