If I know pretty well '(a,b)', I know pretty well 'a', or 'b', or 'a xor b'
There are 4 possibilities, name them e1-e4:

e1  neither match
e2  a only matches
e3  b only matches
e4  both match


Now I restate what you want to prove: Suppose:

\begin{align*}
  \Pr[e4] &amp;\gt 0.25  &amp; \text{prob of correct guess} \\
  \Pr[e2] + \Pr[e4] &amp;\leq 0.5  &amp; \text{prob guess a correctly} \\
  \Pr[e3] + \Pr[e4] &amp;\leq 0.5  &amp; \text{prob guess b correctly}
\end{align*}

This implies that
\begin{align*}
  \Pr[e2] &lt; 0.25  \\
  \Pr[e3] &lt; 0.25  \\
  \implies \Pr[e2] + \Pr[e3] &lt; 0.5  \\
  \implies \Pr[e1] + \Pr[e4] &gt; 0.5,
\end{align*}
and then we notice that this is the probability that the XOR matches.