Converting Prior Knowledge into a Bayesian Prior Suppose I flip a coin 10 times and get 7 heads - based on a Binomial Distribution, I can create a Likelihood Function and estimate the probability of getting a head is 0.7.
Now, suppose someone tells me that they have very strong evidence to believe that the probability of getting a head is 0.9. Suppose I decide to use a Beta Distribution(alpha, beta) as a prior.
My Question: I know that the Expected Value of the Beta Distribution is 'alpha/(alpha + beta)'. If I want to use the Prior Information that was provided to me - does this mean that I have choose values of 'alpha and beta' such that 'alpha/(alpha+beta)' = 0.9? For example - based on the n = 10 samples I observed, in this case, would alpha = 9 and beta = 1?

The typical approach in this case would be to use the conjugate prior (the beta distribution) for the unknown probability parameter and parameterise this using a stipulated prior mean $0 \leqslant \mu_0 \leqslant 1$ and prior effective-sample-size $n_0 \geqslant 0$.  You can then use the reparameterisation of the beta distribution with parameters:
$$\begin{align}
\alpha &= \mu_0 (n_0+2), \\[6pt]
\beta &= (1-\mu_0) (n_0+2). \\[6pt]
\end{align}$$
From the information you have given you would choose $\mu_0 = 0.9$ and you would then have to choose an appropriate value for the prior effective sample-size.  This is a measure of the strength of your prior belief --- the higher the effective-sample-size the stronger your prior belief (and so the lower the prior variance).  You describe the prior belief as "very strong" so you could choose a value like $n_0 = 100$ to represent this, which would mean that you consider your prior belief to be as strong as if it were formed from one-hundred observed data points.
