FYS: Puzzles and Paradoxes--Problem Set 3
Complete exactly three of the following five problems.
1. A man has two children, one of which is a boy born on a Sunday. Assume that the gender and birthday of each child is uniformly and independently distributed.
What are the chances that the other one is a boy? Justify your answer.
2. You and a friend need to decide, fairly, who is going to ask the Professor for a hint to the Bonus Point Problem. Between you, you have but a single coin and you both know that it is biased. Unfortunately, neither of you know what the bias is except that it is not a trick coin, i.e. a coin that always comes up heads or a coin that always comes up tails.
Is it possible for you and your friend to use, via flipping, the biased coin to fairly determine who should ask the Professor for a hint? Justify your answer.
3. Three people sign up to play the following game: The three are led into a room. Once inside, a hat is placed on each of their heads. Each hat has an equal chance of being either white or black. Each person can see the hats that the other two people are wearing, but not his or her own. After they see each other's hats, each person is led away to a separate room. Each person must guess the nature of their own hat, or pass, i.e. say 'My hat is white, 'My hat is black' or 'I pass'. They have no knowledge of what the other players guess or whether they pass. If at least one person guesses correctly, and there are no incorrect guesses (passes are neither correct nor incorrect), they each win a million dollars. If everyone passes, or anyone guesses incorrectly, then they lose and get nothing. They can't communicate in any way between being led into the room and after making their respective guesses. The three can discuss and agree on a strategy before being led into the room.
Is there any strategy that allows the players to win more than 69% of the time? Justify your answer.
4. Three cards are placed inside a hat. One card has a diamond on each side, one card a diamond on one side and a spade on the other, and one card a spade on each side. One card is withdrawn and put on the table with only one side visible. Suppose you see a spade.
If you had to place a bet on what was on the other side, which of the following three options would you choose (assuming you want to win the bet)? Justify your answer.
(i) The other side is a diamond.
(ii) The other side is a spade.
(iii) It does not matter whether I bet (i) or (ii).
5.[Bonus Point Problem] The city of Myriad has 100 building each with 100 stories. It turns out that each building in Myriad is different in the following way--for each building the lowest story from which a dropped egg will break is different. Unfortunately, you know the 'egg-break' story for none of the buildings. Now you just happen to have two very special eggs in your pocket. Your eggs are only ever whole or broken, and don't get cracked or fatigued from multiple droppings, but once broken they cannot be used again. You pick a building at random and you'd like to know exactly what story is the lowest story from which an egg will break if dropped from that building.
What is the lowest number of egg drops that will guarantee you will find the lowest 'egg-break' story? Justify your answer.