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Chapter 15: Q17P (page 743)

Find the number of ways of putting $2$ particles in $4$ boxes according to the three kinds of statistics.

Short Answer

Expert verified

The required values are mentioned below.

$\begin{matrix} M = 16 \\ M = 6 \\ M = 10 \end{matrix}$

Step by step solution

01

Given Information.

The two particles are in four 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

The formula for Maxwell-Boltzmann states that $M = n^{N}$ .

$\begin{matrix} M = 4^{2} \\ = 16 \end{matrix}$

Use Fermi-Dirac method.

$\begin{matrix} M = C (N, n) \\ = C (4, 2) \\ = 6 \end{matrix}$

Use Bose Einstein method.

$\begin{matrix} M = C (n - 1 + N, n) \\ = C (5, 2) \\ = 10 \end{matrix}$

Hence, the required valuesare given below.

$\begin{matrix} M = 16 \\ M = 6 \\ M = 10 \end{matrix}$

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Most popular questions from this chapter

Three coins are tossed; what is the probability that two are heads and one tails? That the first two are heads and the third tails? If at least two are heads, what is the probability that all are heads?

In paying a bill by mail, you want to put your check and the bill (with a returnaddress printed on it) into a window envelope so that the address shows right sideup and is not blocked by the check. If you put check and bill at random into theenvelope, what is the probability that the address shows correctly?

The following measurements of $x$ and $y$ have been made.

$\begin{array}{l} x : 5.1, 4.9, 5.0, 5.2, 4.9, 5.0, 4.8, 5.1 \\ y : 1.03, 1.05, 0.96, 1.00, 1.02, 0.95, 0.99, 1.01, 1.00, 0.99 \end{array}$

Find the mean value and the probable error of $x, y, x + y, x y, x^{3} s i n y$ and. $\ln (x)$

Define s by the equation. $s^{2} = (1 / n) \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}$ Show that the expected valueof. $s^{2} is [(n - 1) / n] σ^{2}$ Hints: Write

$\begin{matrix} {(x_{i} - \bar{x})}^{2} = {[(x_{i} - μ) - (\bar{x} - μ)]}^{2} \\ = {(x_{i} - μ)}^{2} - 2 (x_{i} - μ) {(\bar{x} - μ) + \bar{(x} - μ)}^{2} \end{matrix}$

Find the average value of the first term from the definition of $σ^{2}$ and the average value of the third term from Problem 2. To find the average value of the middle term write

$\begin{matrix} (\bar{x} - μ) = (\frac{x_{1} + x_{2} + \dots + x_{n}}{n} - μ) \\ = \frac{1}{n} [(x_{1} - μ) + (x_{2} - μ) + \dots + (x_{n} - μ)] \end{matrix}$

Show by Problemthat

$\begin{matrix} E [(x_{i} - μ) (x_{j} - μ)] = E (x_{i} - μ) E (x_{j} - μ) \\ = 0 for i \neq j \end{matrix}$

andevaluate $6 .14$ (same as the first term). Collect terms to find

$E (s^{2}) = \frac{n - 1}{n} σ^{2}$

Use Problemto $9$ find the expected value of the sum of the numbers on the dice in Problem $2$ .

See all solutions

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