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3) Imagine you are out playing golf with 2 friends. You need to decide randomly, and fairly, the player to go first. You have a fair coin with you, which you can flip. How do you use the coin to decide who goes first, with an emphasis on minimizing the number of flips you need to make?
You group your friends as if they're one person, so if you lose, you can make another flip for just the both of them.
That doesn't seem fair as you can win on the first flip but it would take one of your friends two flips to win. So chances of winning: You: 50%, Friend1: 25%, Friend 2: 25% The only fair way that I can think of is that you do it like this: 1. You vs Friend 1 2. Winner of #1 vs Friend2 If Winner of #1, wins this then they go first, otherwise 3. Loser of #1 vs Friend2 If Friend2 wins flip #2 and #3 then they go first If everyone wins 1 flip each then you start again. This will happen 25% of the time. When you factor this in the above method finds a winner after 4 flips (3+3/4+3/16+.....) whilst giving everyone an equally fair opportunity to win.
I should clarify: That method finds a winner after 2 flips 50% of the time, after 3 flips 25% and ties after three flips the other 25% of the time. On average it will take 4 flips to win although this will never happen as, if it is tied after 3 flips, then it will take at least 2 more to determine a winner.
With one die, suppose in a round, you earn the amount of dollars equal to the value of the upwards face of the die. Now also suppose after your first roll, you are given the opportunity to cancel your first and roll again, taking that value as the final value. What should your strategy be?
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