You have obtained a patent for a new technology that you are thinking of licensing to a large tech company named Froogle.

Since the technology is fairly premature, it is going to take an extra $3$ million to develop and productize it$.$

Your $($and Froogle$$s$)$ initial assessment is that the market for the technology $($not counting any development costs$)$ is equally likely to be worth $0$M$,2\mathrm{.}5$M$,5$M$,7\mathrm{.}5$M$,$ or $10$M$.$

Assume that Froogle will conduct market research $($which reveals the true potential$)$ only after any licensing deal is signed and before they start development.

a$)$ If you choose to give the technology was away $($for free$)$ to Froogle, what is the expected value to Froogle?

b$)$ If you were to transfer the entire IP to Froogle for a one$-$time fixed payment, what is the maximum that Froogle would be willing to pay?

c$)$ If you were charge a royalty $($as a percentage of final market value$),$ how much will you make if you charged $50\%?$

d$)$ What is the $$optimal$$ royalty that you will like to charge?