Willing to Bet
This game is called "Bob's Lucky Guess!" It's a carnival game where you guess what color marble Bob has drawn out of an opaque bag which contains 5 red marbles and 5 blue marbles. You hand over one dollar to play and then Bob draws a marble out of the bag. If you guess the marble's color, you win two dollars. If you're wrong, you get nothing back. And, in either case, Bob returns the marble to the bag before you play again.
After playing "Bob's Lucky Guess!" for a while and coming out just about even, Bob yawns and proposes a new game. He goes to his marble collection, and adds as many red marbles and as many blue marbles as he wants to the bag. He then comes back to you and tells you that he's changed the count of red and blue marbles but that he won't tell you what changes he's made. He asks you how much you're willing to pay to play the game again, with the same 2 dollar prize if you guesses correctly.
What is true about the amount of money that you should be willing to pay to play the new game?
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1 solution
Regardless of how many red or blue marbles Bob adds to the bag, the strategy of guessing red or blue each with probability 1/2 still guarantees you an expected return of 1 dollar. Depending on the actual distribution of the marbles, you may have a strategy that pays off more. Thus the amount of money you should be willing to bet is at least 1 dollar.