a. The expected value of the gamble is equal to the summation of [rpdoct of the payoffs with their respective probabilities.

So, the expected value of the gamble= (1/3)(45)+(2/3)(0)= $15

b. Certainty equivalent is the amount what an individual is ready to accept over the uncertain higher future value. This means that this is the amount one finds equivalent to the same value as the desirability of the risky asset. So in order to induce an individual to go for a risky asset, one must be compensated by giving them the higher return or simply the premium. When the certainty equivalent is less than the expected value then the person is said to be risk-averse.

c. Since the amount of certain equivalent for the gamble of $8 was less than the expected value of $15 and hence we consider the person as risk-averse.

d. Here the expected value of gamble= (1/3)(-45)+(2/3)(0)= -$15

The amount of certainly equivalent is greater than the expected value of -$15 (which means a high-risk premium of -15-(-8)= -$7) hence it can be said that Lise is risk loving as she's being asked to not play by giving her a higher preimum.