(Solved): please answer the following questions with explanation. Question 1. Consider a game where you can pa ...

Question 1. Consider a game where you can pay either \( \$ 2 \) to flip two coins, or \( \$ 3 \) to flip three coins, and where you win a prize depending on the number of heads that show: - If three heads show, you win \( \$ 7 \). - If two heads show, you win \( \$ 3 \). - If one heads shows, you win \( \$ 2 \). - If no heads show, you don't win anything. (a) Is it better for you to pay \( \$ 2 \) or \( \$ 3 \) (in terms of your expected winnings)? Justify your answer. (b) Suppose that your friend also plays this game, but they randomly pick between paying \( \$ 2 \) or \( \$ 3 \) instead (so there is a \( 50 \% \) probability they flip two coins, and \( 50 \% \) they flip three coins). If you know that they neither won nor lost money (i.e. net winnings is \( \$ 0 \) ), is it more likely that they flipped two coins or three when they played the game? Justify your answer.