You are running a lab which is looking at testing multiple prototypes in parallel. You estimate that the value of each prototype is uncertain, and is equally likely to be $10$ or $0.$ The cost of testing each prototype is $1.$

a$)$ How many prototypes would you test?

b$)$ Draw the curve of expected value from testing k prototypes as a function of k$.$

c$)$ How does this curve change if the standard deviation $($of each prototype$$s value$)$ is higher?

d$)$ How does this curve change if the mean is higher?

e$)$ We assumed that the values are independently distributed $($i$.$e$.,$ the potential correlation between the values is zero$).$ How does the curve change if the values are correlated?