Omega Company has developed a new gadget. If the gadget is successful, the present value of the payoff $($at the time the product is brought to market$)$ is $$21$ million. If the gadget fails, the present value of the payoff is $$8$ million. If the gadget goes directly to market, there is a $50$ percent chance of success. Alternatively, the company can delay the launch by one year and spend $$1\mathrm{.}10$ million to test$-$market the gadget. Test$-$marketing would allow the firm to improve the gadget and increase the probability of success to $75\%.$ The appropriate discount rate is $12\%.$ Should the firm conduct test$-$marketing?

No$,$ because NPV is lower by $$237,588$

No$,$ because NPV is lower by $$188,974$

Yes, because NPV is higher by $$469,862$

D$.$ Yes, because NPV is higher by $$377,850$

E$.$ Yes, because NPV is higher by $$248,214$