Game 2
1 - It still doesn't matter which color I choose. Since I can't know the proportions, it's still identical odds.
2 - Again, same bet. If the odds of picking my color are the same, and the prize is the same, the bet needs to be the same.


But I'm really interested in hearing where I failed. I probably did, so I'm eager to learn something new.

I'm not sure where Naj is going with this, but your analysis for Game 2 isn't exactly right. That's why I asked him the question about the composition of the balls. In Game 2, the distribution of balls is neither evenly split nor random. Thus, Naj has the ability to skew the proportion of balls in the jar - he could fill it entirely with red, or black, balls if he chose.

Thus, Naj has the ability to convert the game from a game of chance into a guessing game (in whole or in part). To use a simpler case, suppose I write down "heads" or "tails" on an envelope, and then flip a fair coin to see what happens (I win on a match). That's a fifty-fifty proposition. But suppose I write down my choice in an envelope, and then ask Naj to guess what I wrote. Now it's no longer a game of pure chance - it's affected by whether Naj has any information (however imperfect) about my predilection to choose either heads or tails. I can also engage in second-order guessing, skewing my own answer to counter what I think Naj might choose.

I'm not sure how big that effect might be, but it does change the odds from pure 50/50 by introducing a non-random element.

Albaby