Utility of expected income or expected utility of income?
Your problem has problems with it.  There is no disutility of studying and there is no interest rate or it is zero percent.  However, let's consider the three possible cases.

If $y_2&gt;F$ then for all cases you should study.  If $y_1&lt;F$ then all cases you should not study.  The difficulty happens when $y_2&lt;F$ and $y_1&gt;F$.  Then you should study if $E(U(s))\gg{U}(F)$ and not if $E(U(S))\ll{U}(F)$.  The difficulty with the phrasing of the problem arises when $E(U(s))\approx{U}(F)$.

From the form of your problem, indifference should happen at: $$(y_1^\alpha-y_2^\alpha)p+y_2^\alpha=F^\alpha.$$  It should have been written as: $$(y_1^\alpha-y_2^\alpha)p+y_2^\alpha-U(s=study)=F^\alpha(1+\bar{r})^\alpha.$$  You cannot differentiate over $s$ because it is a binary choice.  In fact, $y_1,y_2$ are really $y_1(s=study)$ and $y_2(s=study)$ and $F$ is really $F(s=\text{don't study})$.

For the form of your problem, you should be indifferent if: $$p=\frac{F^\alpha-y_2^\alpha}{y_1^\alpha-y_2^\alpha}.$$  This permits an objective solution that does not depend upon the person.  For the more expansive problem at the indifference point, the solution depends upon the market rate of interest and the subjective cost of studying.

This implies two important things.  The first is that the value of an education depends on market rates of interest.  Because of this, there should be a decrease in the rate of education when rates are high.  The second is that the indifference point for the probability depends on the specific person.  This implies that probability has a subjective component, even when the probability is external to the person.  Indeed, if the probability is a function of all the choices of all the people, then the probability actually depends upon the marginal rate of disutility experienced in studying.