Remox Corporation is considering a short production run for either of two new products: a temperature sensor or a pressure sensor$.$ Developing the temperature sensor will cost $$100,000,$ while developing the pressure sensor costs only $$10,000.$ The risk is that these investments may not result in viable products. In particular, there is a $0\mathrm{.}5$ probability of success in developing the temperature sensor$,$ and a $0\mathrm{.}8$ probability of success in developing the pressure sensor$.$ The market for each product is known Figure $1$: Decision tree for Remox Corporation without informationif the products can be developed successfully: selling the temperature sensors could generate revenue of $$1$ million and selling the pressure sensor could generate revenue of $$400,000.$ Alternatively they could opt to do nothing, in which case they would not incur any development costs, nor would they earn any revenues from product sales.

Using the information above, fill in the blanks in the decision tree in Figure $1$ below.

$($a$)$ What is the expected value of choosing to develop the temperature sensor$?($b$)$ What is the expected value of choosing to develop the pressure sensor$?$

$($c$)$ If you were the manager of Remox Corporation, would you choose to develop either of these two products? If so$,$ which product would you choose to develop?