Working out various probabilities using a model for latent variable 

I'm really not sure where to start with this so if any advice could be given I would greatly appreciate it. 
 A: Notice that $y^\star_i$ follows a Normal law
$$y^\star_i\sim N(\alpha+\beta x^\prime_i,1)$$
(see https://en.wikipedia.org/wiki/Normal_distribution properties)
Then simply write down things:
$$
Pr(y_i=0) \Leftrightarrow Pr(y^\star_i<\gamma_1) \Leftrightarrow  \Pr(\alpha+\beta x^\prime_i<\gamma_1) \Leftrightarrow  Pr(x^\prime_i<\frac{\gamma_1-\alpha}{\beta}) = \Phi(\frac{\gamma_1-\alpha}{\beta})
$$
where $\Phi$ is the cumulative function (also read about Error function).
For (iii) simply use that fact that probability sum to one (I)+(II)+(III)=1
I think it is a good start.

Update reading some Dimitriy meaningful comments I realized that my explanation is not clear at all. Below I will try to clarify this. 
Dimitriy suggests (tell me if I do not understand you)
$$
u_i=y_i^\star-\alpha-\beta x_i^\prime
$$
thus
$$
y_i^\star<\gamma_1 \Leftrightarrow u_i<\gamma_1-\alpha-\beta x_i^\prime
$$
then 
$$
Pr(y_i^\star<\gamma_1)=Pr(u_i<\gamma_1-\alpha-\beta x_i^\prime)=\Phi_1(\gamma_1-\alpha-\beta x_i^\prime)
$$ 
with $\Phi_1$ the Cumulative distribution of $N(0,1)$. In full details we get:
$$
Pr(y_i^\star<\gamma_1)=\frac{1}{2} \text{erfc}\left(\frac{\alpha -\gamma_1 +\beta 
   x_i^\prime}{\sqrt{2}}\right)
$$
(where $\text{erfc}$ is the complementaey error function )
My clarified explanation:
If $u_i$ is a random variable then $y^\star_i=\alpha+\beta x^\prime_i+u_i$ is also a random variable. A random variable is nothing more than a (measurable) function (on proper spaces). In our peculiar case we have $u_i\sim N(0,1)$, thus:
$$y^\star_i\sim N(\alpha+\beta x^\prime_i,1)$$
then we only have to write:
$$
Pr(y^\star_i<\gamma_1)=\Phi_2(\gamma_1)
$$
but this time $\Phi_2$ is the cummulative distribution of $N(\alpha+\beta x^\prime_i,1)$. In full detail we get
$$
Pr(y_i^\star<\gamma_1)=\frac{1}{2} \text{erfc}\left(\frac{\alpha -\gamma_1 +\beta 
   x_i^\prime}{\sqrt{2}}\right)
$$
which is hopefully identical to Dimitriy solution.
What I did wrong:
In my initial post, after having written $Pr(y_i^\star<\gamma_1)$ I decided (for a reason I ignore) to express this in terms of $x^\prime_i$:
$$
Pr(x^\prime_i<\frac{\gamma_1-\alpha}{\beta}) = \Phi_3(\frac{\gamma_1-\alpha}{\beta})
$$
But now the $\Phi_3$ is the CDF of $x^\prime_i$ considered as a random variable. From $x^\prime_i=\frac{y_i^\star-\alpha-u}{\beta}$ we have 
$$
x^\prime_i\sim N(\frac{y_i^\star-\alpha}{\beta},\frac{1}{|\beta|})
$$
and
$$
Pr(x^\prime_i<\frac{\gamma_1-\alpha}{\beta}) = \frac{1}{2} \text{erfc}\left(\frac{\text{sign}(\beta)  (y^\star_i-\gamma_1
   )}{\sqrt{2} }\right)
$$
I think this is correct but, compared to the previous solution, it gives the probability in terms of $y^\star_i$ and not in term of $x^\prime_i$. In all cases this was not clear at all in my explanation, I am sorry for that (because that was not a good start!) and I thank Dimitriy for his initial comment.
