I am confused on how to calculate the beta in a probit/logit model
Probit Model $P(Y_i=1)=\Phi(X'\beta)$
Logit Model: $P(Y_i=1)=e^{X'\beta}/(1+e^{X'\beta})$
These formula's are great but how do I calculate the beta's in the model? What is a proper estimator for beta hat?
It is hard to answer your question, as you do not provide the background you have. Here is a general answer.
You want to find $\mathbb{P}[Y_i = 1 | X_i]$ when you have a bunch of data that comes from the same distribution, that is, the result ($Y_i$) is influenced the same way by the inputs ($X_i$) of each sample you have. Read more: I.I.D. (Wikipedia).
You know $X_i$ and $Y_i$ for your samples, and you want to find a function that maps the inputs to the outputs as close as possible. This is what the probit/logit models are doing, and they have a parameter $\beta$ to dictate the influence of the different input variables. To find the best $\beta$, we have first to define what the best is.
In linear regression models, the best model is often defined by the model that has the best $R^2$ measure with the output, or the one that has the smallest Mean Squared Error (Wikipedia). In classification tasks, a common measure is the Logistic loss (Wikipedia).
The value of the loss function is $L(y, f(x,\beta))$ where $L$ is the loss function, $x,y$ the input and output variables, and $\beta$ are the parameters of your model. It is a comparison of your model with the real result $y$, and gives you a metric to evaluate your model. You want to have the lowest error with respect to $\beta$, $$\hat{\beta} = \arg \min_{\beta} L(y,f(x,\beta))$$ Depending on your model and cost function, you can equate the derivative to 0 and compute the $\beta$ that makes it possible based on the data. In the case of the transformed models you are interested about, this is not possible, as as mentionned in the comments, no closed solution exists. However, you can come very close to the optimal solution by using gradient descent (Wikipedia). The idea is that the derivative of the function at a certain $\beta$ will point to the direction of highest increase in the cost function with respect to $\beta$, and if you go in the other direction, your $\beta$ will improve and the error will lower. As long as your cost function is convex, you will fill a global minimum, the optimal $\beta$.
How to find the parameters of the logistic regression is the introduction of a lot of books and classes in Machine learning. For a more specific answer, I'd advise looking at
You can also take a look at questions on a similar topic on CrossValidated,
