Question

Do Problem $15$for $2$particles 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" $\frac{1}{3}$, that is, they are equally probable.)

Short answer

The required values are mentioned below.

$\begin{array}{c}P\left(B_1\right)=P\left(B_2\right)\\ =P\left(B_3\right)\\ =\frac{1}{3}\end{array}$

Step-by-step solution

Given Information

The two particles are in two boxes.

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.

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.

$\begin{array}{c}M=C(n-1+N,N)\\ =C(2-1+2,2)\\ =C(3,2)\\ =3\end{array}$

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

$\begin{array}{c}P\left(B_1\right)=P\left(B_2\right)\\ =P\left(B_3\right)\\ =\frac{1}{3}\end{array}$

Hence, the required valuesare given below.

$\begin{array}{c}P\left(B_1\right)=P\left(B_2\right)\\ =P\left(B_3\right)\\ =\frac{1}{3}\end{array}$