A leading pharmaceutical company, BioGenetix Innovations, are at the forefront of developing a new

treatment for a rare neurological disorder. BioGenetix have identified two chemical compounds for

further development, but it only has the resources to proceed to clinical trials with only one of these

compounds. The compounds under consideration are NeuroCure$-$A$,$ synthesised in their laboratory in

Geneva, and SynapseTherapy$-$B$,$ developed at their research facility in Cambridge.

Initial laboratory tests on NeuroCure$-$A show a $70\%$ chance of it being highly effective with minimal side

effects, which would result in development and regulatory approval costs of $50$ million. However, if

further testing reveals significant side effects, the costs could rise to $90$ million due to additional

research and chemical modifications as well as extended regulatory scrutiny.

The alternative compound, SynapseTherapy$-$B$,$ has shown consistent results, with a high likelihood of

effectiveness as well as manageable side effects. The development and approval costs for

SynapseTherapy $B$ are estimated to be $70$ million.

Furthermore, BioGenetix Innovations is considering whether to commission a renowned university in

London to conduct a more detailed analysis of NeuroCure$-$A$\text{'}$s side effects. However, the university has

explained that its analysis offers an $80\%$ probability of giving a correct indication of the nature of the

compound's side effects and whether they are significant or minimal.

$($a$)$ BioGenetix Innovation's Finance Director would like to minimise expected costs. What is the

expected value of the imperfect information from the London university's detailed analysis? How much

should BioGenetix pay the university?

$($b$)$ The London university has just made significant advances in its analysis of chemical compounds, and

they can now offer perfect information, giving a certain indication of whether NeuroCure$-$A will be

effective with minimal side effects. What is the value of the information from the university's analysis?$|$