Entanglement, projection operator and partial trace

[Matt atkinson]

Homework Statement

Consider the following experiment: Alice and Bob each blindly draw a marble from a vase that contains one black and one white marble. Let’s call the state of the write marble $|0〉$ and the state of the black marble $|1〉$.
Consider what the state of Bob’s marble is when Alice finds a white marble

Homework Equations

The Attempt at a Solution

So I found the mixed state of Bob and alice's particle to be:
$$\rho=\frac{1}{2}|0,1\rangle \langle0,1|+\frac{1}{2}|1,0\rangle \langle1,0|$$
And i know that finding a white marble can be described in the following way:
$$\rho^B=\frac{Tr_A(|0\rangle_A\langle0|\rho)}{Tr(|0\rangle_A\langle0|\rho)}$$
where $Tr_A$ is the partial trace w.r.t Alice's system.
And just by reasoning i know the answer should be $|1\rangle\langle1|$ but I am struggling to prove that by solving the above equation.
Here's my attempt:
$$\rho^B=\frac{Tr_A(|0\rangle_A\langle0|(|0,1\rangle \langle0,1|+|1,0\rangle \langle1,0|))}{Tr(|0\rangle_A\langle0|(|0,1\rangle \langle0,1|+|1,0\rangle \langle1,0|))}$$
$$\rho^B=\frac{Tr_A(|0\rangle_A\langle0|(|0\rangle \langle0| \otimes |1\rangle \langle1|+|1\rangle \langle1|\otimes|0\rangle \langle0|))}{Tr(|0\rangle_A\langle0|(|0\rangle \langle0| \otimes |1\rangle \langle1|+|1\rangle \langle1|\otimes|0\rangle \langle0|))}$$
But I am not quite sure where to go from there, I am a little inexperienced using Braket notation so any pointers would be greatly appreciated.

 

[Asim Riaz]

A:If Alice finds a white marble, this means that the two-particle state in Bob's vase is $|0\rangle_A\otimes |1\rangle_B$. Tracing out Alice's system just gives you the state of Bob's marble, which is $|1\rangle_B \langle 1|$.