I think "speaks the truth two days a week alternately" means he tells the truth two days a week, with one day in-between when he lies. If he simply alternates, why would "two" be in there? Also, if he never tells the truth two days in a row or lies two days in a row, then he can't tell the truth the same days each week.
If the last statement is false, he must tell the truth on Wed and Fri. That makes the first statement true, since he only tells the truth twice per week. So the statements must be TFF. This can only be the case for Fri-Sat-Sun, but that would make the second statement true. Ergo, we must conclude the last statement is true.
The options now are:
either TFT or FFT. If it's TFT, he lies on Mon and Tue, so the first statement must be Wed, Thu, or Fri (all other options have him tell the truth on Mon or Tue). If the first statement was made on Wed, that would make Wed/Fri the truthful days, and the final statement false. If it's Thu, that works out. If it's Fri, then the second statement was made on Sat, and would have been true. Thus, one option is that the truthful days are Thu/Sat, the statements are made Thu-Fri-Sat, and are TFT.
On the other hand,
if the statement truth-values are FFT, then he must tell the truth on one of Mon or Tue. The four options are: Truthful on Sat/Mon, statements on Thu/Fri/Sat, Truthful on Sun/Tue, statements on Fri/Sat/Sun, Truthful on Mon/Wed, statements on Sat/Sun/Mon, Truthful on Tue/Thu, statements on Sun/Mon/Tue. Of these options, the second and third do not work, because the second statement would be true. The others do work.
So the options remaining are:
Truthful on Thu/Sat, Spoke on Thu/Fri/Sat, truth values TFT, truthful on Sat/Mon, spoke on Thu/Fri/Sat, truth values FFT, or truthful on Tue/Thu, spoke on Sun/Mon/Tue, truth values FFT.
If each case is equally likely,
Then the odds are 2/3 that he speaks the truth on Thursdays.