Your Game 2 is poorly defined. What does it mean to &quot;guess correctly&quot; when you&#x27;re asked to guess twice? If you systematically take the first (or the last) answer, then it&#x27;s equivalent to being awaken only once on Tails. If you take the logical OR of the correctness of two guesses, then the sleeping beauty can win 5&#x2F;8 of the time by answering randomly, which is better than 1&#x2F;2 by guessing a fixed answer (the only strategy you analyzed).