Problem Set 4

Probabilities and Counting States

Problem Set 4: Probabilities and Counting States

1. A Deck of Cards

Consider a standard deck of playing cards. There are 52 cards in total. These cards are

divided among 4 suits (clubs, diamonds, hearts, spades). And within each suit, there are 13

types of cards (ace, 2, 3,...,10, jack, queen, king). Suppose you are dealt a 5-card hand from

a full, well-shuffled deck.

a How many different hands are possible? (Here, the ordering of the cards within the hand

is not important.)

b What is the probability of being dealt a flush (all 5 cards have the same suit)?

c What is the probability of being dealt one pair (two cards of the same type) and only one

pair?

d What is the probability of not getting one pair (and only one pair) until the nth hand?

Here, each successive hand is dealt from a full, well-shuffled deck.

2. The Russian Roulette

In the game of Russian roulette ( Do not try to play it at home without adult supervi-

sion), one inserts a single cartridge into the drum of a revolver, leaving the other five

chambers of the drum empty. One then spins the drum, aims at one’s head, and pulls

the trigger. For now, assume that you are playing it alone, out of sheer depression.

a What is the probability of you being still alive after playing the game N times?

b What is the probability of surviving (N − 1) turns in this game and then being

shot the Nth time you pull the trigger?

c What is the mean number of times you get the opportunity of pulling the trigger in

this game?

dNow Suppose you play this game with your best friend. What is the probabil-

ity of you losing your friend if you take the first turn? if your friend takes the first turn?

Remember that for all of above scenarios, the drum is spun after every turn.

Due date: April 6, 2010, 11:30 am 1


3. A Random Walk

A drunk starts out from a lamppost in the middle of a street, taking steps of equal

length either to the right or to the left with equal probability. What is the probability

that the man will again be at the lamppost after taking N steps

a if N is even?

b if N is odd?

4. More Random Walk

Consider the random walk of the drunk again but this time assume that he is twice

as likely to take a step towards right (where the pub is ) than to the left. On average,

what is the net displacement to the right after N steps?

5. Gas in a Box

Consider a gas of N0 noninteracting molecules enclosed in a container of volume V0.

Focus attention on any subvolume V of this container and denote by N the number of

molecules located within this subvolume. Each molecule is equally likely to be located

anywhere within the container; hence the probability that a given molecule is located

within the subvolume V is simply equal to VV0.

a What is the mean number < N > of molecules located within V ? Express your

answer in terms of N0, V0 and V .

b Find the relative dispersion ( a measure of fluctuations away from the mean number)

<(N−<N>)2><N>2 in the number of molecules located within V . Express your answer in

terms of N , V , and V0.

c What does the answer to part (b) become when V ≪0

d What value should the dispersion found in part (b) assume when V → V0?

6. Angular Momentum

In a certain quantum mechanical system the x component of the angular momentum,

Lx, is quantized and can take on only the three values −~, 0, or ~. For a given state

of the system it is known that < Lx >= 13~ and < L2

x >= 23~2. Here < Lx > is the



mean of Lx and < L2x > is the mean of the square of Lx while ~ is a constant. Find

the probability for each of the Lx values to be found in a random measurement.

7. Electron energy

The probability density ρ(E) of finding an electron with energy E in a certain system

is given by

ρ(E) =0.2δ(E + E0) for E < 0 (1)

=0.8(1

b)e−

Eb for E < 0 (2)

where E0 = 1.5eV and b = 1.0eV .

a What is the probability that E > 0?

b What is the mean energy of electron?

8. The Photons

Consider the Bose-Einstein density for finding n number of electrons in a given mode

of black body radiation

ρ(n) = (1− a)an n = 0, 1, 2 . . . , a < 1 (3)

Find the mean number of photons in this mode < n > and the variance < n2 > − <

n >2.

9. 2-Dimensional Gas

Consider N particles living on a 2 dimensional surface. The particles move freely except

for the fact that they are confined to an area A. The particles have total energy equal

to E. Calculate the phase space volume Ω(E) of allowed states in micro-canonical

ensemble and find :

a) What is the energy in terms of temperature? How does it compare to 3-dimensional

ideal gas?

b) What is the equation relating “Pressure”, “Area” and temperature? Note that in

this case pressure is defined as force per unit length and work done on the gas is given



by

dW = −pdA

.

10. Balls in large dimensions

Starting with the expression for w(E), the volume of phase space with energies less

than or equal to E, for an ideal gas, show that the entropy calculated with

S = k log Ω(E)

is the same as

S = k logw(E)

in the limit of very large N . Here Ω(E) is the volume of a thin shell with energies

between E and E +∆, and is given by

Ω(E) =dw(E)

dE∆

.

Copy the form of w(E) for the ideal gas from lecture notes.

11. A System of Classical Harmonic Oscillators

We can model a solid as a collection of harmonic oscillators, each atom in the crystal

lattice acting as an oscillator, vibrating around its mean position. Consider a very

simplified model of a solid, consisting of N oscillators all having same frequency w

and all are independent of each other. Each oscillator can vibrate in three directions,

again with same frequency. The total energy of the system is therefore given by, in

terms of the coordinates and momenta of each oscillator,

H(p, q) =3N∑i=1

(p2i2m

+1

2mw2x2

i

)



a)Calculate the number of allowed micro states Ω(E,N), if this system has total energy

equal to E. (Hint: You have to calculate a very similar integral as in case of ideal gas.

This time you can not integrate position coordinates trivially to give physical volume.

Change variables so that the integral becomes a volume of 6N dimensional ball.

b) Taking the formula for entropy to be S = k log Ω, find out an equation of state

relating energy and temperature. Find also the heat capacity C = dEdT. How does it

differ from heat capacity of ideal gasses.


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Problem Set 4