Its an interesting to which the answer is be no means too obvious. It is, I hope, clear that if you played the game sufficient times the average return per game would be the same whether you always switched or never switched. Equally it is clear that you could write two cheques like that and make the offer. If is also true that on an individual game by choosing the face-down check instead of the one you've revealed, you stand to gain K dollars at the risk of losing only K/2. So the flaw must be in the assertion that it is therefore sensible to switch. And it is, since we agreed that overall switching brings no benefit. SO we have a mismatch in our understanding of potential benefit for a single game and the impact that has on out our probable benefit.