Clover Park Racing Team has developed a new Formula One car, "Velocity X$"($Model # $4065),$ which

claims to have groundbreaking aerodynamics and fuel efficiency, potentially revolutionizing the racing

experience. "VelocityX" has not yet demonstrated its performance in official races and is undergoing

phase I testing to evaluate the efficacy of its new design. Despite the lack of racing data to support the

claims of Clover Park Racing, the prominent racing company, Toble Motorsports, is interested in

acquiring the rights to manufacture and distribute model #$4065$ provided it passes three sequential stages

of performance testing.

The development of a new Formula One car is often a long and failure$-$ridden process. Typically, it takes

an average of $10-15$ years to bring a car model from initial design to the racing track. On average, for

every model that reaches competitive racing, $5,000$ to $10,000$ designs are explored in the development

phase.

Toble Motorsports, early this month, approached Clover Park Racing with an offer to purchase the rights

to Model #$4065$ for $$30$ million. If "VelocityX" manages to pass all three phases of performance testing,

the commercialization of "VelocityX" would be very lucrative for Toble Motorsports. If "VelocityX"

could be marketed as having superior aerodynamics and fuel efficiency, it is forecasted that the model

would have a market value of $$510$ million $($this and all subsequent market values are stated at the present

value$).$ If "VelocityX" was approved for superior aerodynamics only, the present market value would be

$$430$ million, and $$50$ million for superior fuel efficiency only.

Phase I of performance testing for Model #$4065$ has been underway for nearly $10$ months. To date, there

have been no critical failures or safety issues in the test runs with a team of $150$ skilled drivers. Based on

the development notes provided by Clover Park Racing, and Toble Motorsports' experience with the

testing and development process, Toble Motorsports estimates that Model #$4065$ has a $60\%$ chance of

successfully completing Phase I testing and $40\%$ chance of failing phase I testing. If Toble Motorsports

were to purchase the rights to Model #$4065,$ the cost to the company would be an additional $$5$ million to

complete phase I testing.

If Model #$4065$ were to pass Phase I, the next step would be to proceed to Phase II testing. If Model

#$4065$ were to pass Phase II$,$ the next step would be to proceed to Phase III testing.

Phase II testing involves rigorous track testing with several hundred skilled drivers to determine the

model's performance in aerodynamics and fuel efficiency as well as to document any mechanical issues.

Tobel Motorsports estimates that there is a $10\%$ chance that Model #$4065$ would show superior

performance in aerodynamics only, a $10\%$ of superior fuel efficiency only, and a $30\%$ chance of excelling

in both areas simultaneously. Phase II of testing would require two years to complete and would cost $$40$

million.

Phase III of testing requires that the car be tested in multiple racing scenarios over a multi$-$year time

horizon. The cost and success of Phase III testing depends on the results of Phase II testing. If Phase II

testing indicated that Model #$4065$ excelled in aerodynamics only, the Phase III test would cost $$50$

million and have a $75\%$ chance of meeting international racing standards. If it is excellent in fuel

efficiency only, the Phase III test would also cost $$50$ million and have a $70\%$ chance of meeting

international racing standards. However, if Phase II testing showed superiority in both areas, the required

Phase III testing would cost $$140$ million and have a $60\%$ chance of meeting international racing

standards sufficient to demonstrate superiority in both aerodynamics and fuel efficiency, a $15\%$ chance

for aerodynamics only, and a $10\%$ chance for fuel efficiency only.

a$)$ Draw the corresponding decision tree for the above scenario. $(20$ Points$)$

b$)$ From the perspective of Toble Motorsports, what is the expected payoff of the decision whether

to purchase the rights to "VelocityX" $($i$.$e$.,$ Model #$4065)?$ Show your work. $(10$ points$)$

c$)$ If you were the CEO of Toble Motorsports, would you recommend purchasing the rights to

"VelocityX" $($i$.$e$.,$ Model #$4065)?$ Why or why not? $(3$ points$)$