Four games, one winner. Below are four versions of the same game. Your archnemisis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs $1 to play each game. Explain your reasoning.
(a) If the proportion of heads is larger than 0.60, you win $1.
(b) If the proportion of heads is larger than 0.40, you win $1.
(c) If the proportion of heads is between 0.40 and 0.60, you win $1.
(d) If the proportion of heads is smaller than 0.30, you win $1.