I'd like to verify my solution and receive wider clarification where it's possible. This is an exercise from Bayes Rules! book.
Exercise 3.1 (Tune your Beta prior: Take I) In each situation below, tune a Beta(α,β ) model that accurately reflects the given prior information. In many cases, there’s no single “right” answer, but rather multiple “reasonable” answers.
a. Your friend applied to a job and tells you: “I think I have a 40% chance of getting the job, but I’m pretty unsure.” When pressed further, they put their chances between 20% and 60%.
First of all, I think that "a 40% chance of getting the job" should be treated as mode of the beta distribution, because according to the context 40% is the most probable value where pdf respectively reaches its maximum. Playing with numbers around using the formula $$\frac{\alpha-1}{\alpha+\beta-2}$$
I found reasonable $\alpha=11$ and $\beta=16$
b. A scientist has created a new test for a rare disease. They expect that the test is accurate 80% of the time with a variance of 0.05.
In this case I think that 80% must be treated as mean of the beta. "A variance of 0.05" is actually a variability and therefore must be treated as standard deviation of the beta. In fact I raised an issue about the misleading formulation https://github.com/bayes-rules/bayesrules/issues/87. Using the formula for the mean of the beta distribution $$\frac{\alpha}{\alpha+\beta}$$ it's obvious that $\beta=\frac{1}{4}\alpha$
Trying various numbers, I stopped on $\alpha=48$ and $beta=12$. With these parameters $SD(x) = 0.05121475$ which is close to "a variance of 0.05".
c. Your Aunt Jo is a successful mushroom hunter. She boasts: “I expect to find enough mushrooms to feed myself and my co-workers at the auto-repair shop 90% of the time, but if I had to give you a likely range it would be between 85% and 100% of the time.”
"90% of the time" is mean of the beta. So $\beta=\frac{1}{9}\alpha$. I found the reasonable values of $\alpha=90$ and $beta=10$. The beta cdf(0.85) with these parameters gives us 0.05952991 which I consider fitting for "likely range it would be between 85% and 100% of the time".
d. Sal (who is a touch hyperbolic) just interviewed for a job, and doesn’t know how to describe their chances of getting an offer. They say “I couldn’t read my interviewer’s expression! I either really impressed them and they are absolutely going to hire me, or I made a terrible impression and they are burning my resumé as we speak.”
This is the most interesting case in my opinion. As all the events are equally possible then it leads to the uniform probability distribution with $\alpha=1$ and $\beta=1$.
