A measurement can be associated with a decomposition of the identity $I$ into a sum of projectors. For example, the measurement of a qubit in the $| + \rangle ,| - \rangle$ basis is associated with the decomposition
$$
I = P_+ + P_- = |+\rangle\langle+| ~+~ |-\rangle\langle-|.
$$
Then the probabilities are $\text{Tr}(P_+|\phi\rangle\langle\phi|) = |\langle\phi|+ \rangle|^2$ and $\text{Tr}(P_-|\phi\rangle\langle\phi|) = |\langle\phi|- \rangle|^2$ if we measure a qubit state $|\phi\rangle$.
When we measure only one subsystem this is equivalent to a measurement of the whole system with a naturally extended decomposition. For example, in your case it will be
$$
I_{123} = I_{12} \otimes P_+ + I_{12} \otimes P_-,
$$
where $I_{123}$ is the identity operator on the whole system of three qubits, and $I_{12}$ is identity on just first two qubits.
Then the probabilities are $p_+ = \text{Tr}((I_{12} \otimes P_+)|\gamma\rangle\langle\gamma|)$ with the post-measurement state $\frac{1}{\sqrt{p_+} } (I_{12} \otimes P_+) |\gamma\rangle $ and $p_- = \text{Tr}((I_{12} \otimes P_-)|\gamma\rangle\langle\gamma|)$ with the post-measurement state $\frac{1}{\sqrt{p_-} } (I_{12} \otimes P_-) |\gamma\rangle $.
If you are not interested in post-measurement states then there is another approach of calculating probabilities $p_+,p_-$. You can calculate the density matrix of the third qubit
$$
\rho_3 = \text{Tr}_{12} (|\gamma\rangle\langle\gamma|) = \text{Tr}_{12}\big(\frac{1}{2}(| 010 \rangle + | 101 \rangle ) (\langle 010 | + \langle 101 | )\big)
$$
$$
= \text{Tr}_{12}\big(\frac{1}{2}(|010\rangle\langle 010| + |010\rangle\langle 101| + |101\rangle\langle 010| + |101\rangle\langle 101| )\big) =
$$
$$
= \frac{1}{2} (|0\rangle\langle 0| + |1\rangle\langle 1|) = \frac{1}{2} I.
$$
Then the probabilities are $p_+ = \text{Tr}(P_+ \rho_3) = \text{Tr}(P_+ \frac{1}{2} I) = \frac{1}{2}$ and $p_- = \text{Tr}(P_- \rho_3) = \text{Tr}(P_- \frac{1}{2} I) = \frac{1}{2}$.
The post-measurement states will actually be
$$
\frac{1}{\sqrt{p_+} } (I_{12} \otimes P_+) |\gamma\rangle =
(I_{12} \otimes |+\rangle\langle+|)(|010\rangle + |101\rangle) =
\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)|+\rangle
$$
and
$$
\frac{1}{\sqrt{p_-} } (I_{12} \otimes P_-) |\gamma\rangle =
(I_{12} \otimes |-\rangle\langle-|)(|010\rangle + |101\rangle) =
\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)|-\rangle
$$
