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Chapter 15: Q16P (page 743)

Do Problem 15for 2particles in 2 boxes. Using the model discussed in Example role="math" localid="1654939679672" 5, find the probability of each of the three sample points in the Bose-Einstein case. (You should find that each has probabilityrole="math" localid="1654939665414" 13, that is, they are equally probable.)

Short Answer

Expert verified

The required values are mentioned below.

PB1=PB2=PB3=13

Step by step solution

01

Given Information

The two particles are in two boxes.

02

Definition of uniform sample spaces.

If a given experiment's sample space is known to be uniform, the probability of an event can be calculated using the event sizes and the sample space.

03

Find the methods.

Use the Bose Einstein method. The number of method particles are not distinguishable but allow to put 2 balls in the same box. The number of methods arranged is given below.

M=C(n−1+N,N)=C(2−1+2,2)=C(3,2)=3

All cases probably have three cases so the probability of each case is given below.

PB1=PB2=PB3=13

Hence, the required valuesare given below.

PB1=PB2=PB3=13

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Most popular questions from this chapter

Equation (10.12)isonly an approximation (but usually satisfactory). Show, however, that if you keep the second order termsin,(10.10)then

role="math" localid="1664364127028" w¯=w(x¯,y¯)+12(∂2w∂x2)σx2+12(∂2w∂y2)σy2.

(a) Suppose you have two quarters and a dime in your left pocket and two dimes and three quarters in your right pocket. You select a pocket at random and from it a coin at random. What is the probability that it is a dime? (b) Let x be the amount of money you select. Find E(x).

(c) Suppose you selected a dime in (a). What is the probability that it came from your right pocket?

(d) Suppose you do not replace the dime, but select another coin which is also a dime. What is the probability that this second coin came from your right pocket?

Suppose13people want to schedule a regular meeting one evening a week. What is the probability that there is an evening when everyone is free if each person is already busy one evening a week?

The following problem arises in quantum mechanics (see Chapter 13, Problem7.21). Find the number of ordered triples of nonnegative integers a, b, c whose suma+b+c is a given positive integer n. (For example, ifn=2, we could have(a,b,c)=(2,0,0)or(2,0,2)or (0,0,2)or (0,1,1)or or (1,0,1)or (1,1,0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in3boxes, and follow the method of the diagram in Example 5.

Define s by the equation.s2=(1/n)∑i=1n(xi−x¯)2Show that the expected valueof.s2is[(n−1)/n]σ2Hints: Write

(xi−x¯)2=[(xi−μ)−(x¯−μ)]2=(xi−μ)2−2(xi−μ)(x¯−μ)+(x¯−μ)2

Find the average value of the first term from the definition ofσ2and the average value of the third term from Problem 2. To find the average value of the middle term write

(x¯−μ)=(x1+x2+⋯+xnn−μ)=1n[(x1−μ)+(x2−μ)+⋯+(xn−μ)]

Show by Problemthat

E[(xi−μ)(xj−μ)]=E(xi−μ)E(xj−μ)=0   fori≠j

andevaluate6.14 (same as the first term). Collect terms to find

E(s2)=n−1nσ2
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