Question: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a sports car; behind all of the others is bicycles. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 2, revealing a bicycle. He then says to you, "Do you want to pick door No. 3?" Is it to your advantage to switch your choice?
Answer: Yes, by changing your answer your chances of winning actually goes up from 1/3 to 2/3.
This becomes obvious when expanding the example. Suppose there was 100 doors rather than 3. You pick one and the host shows you that the car is not behind 98 of the doors then asks you to switch to the remaining door or keep the door you picked. Of course you would switch your door because chances are you didn't pick the correct door initially.
For more explanation go to http://en.wikipedia.org/wiki/Monty_Hall_problem#Solutions
Question: In the harbor of a small town, there is a large boat. Coming off of the boat is a ladder reaching down so that the bottom rung is just above the surface of the water. As the tide comes in, the water rises at 1 foot every 30 minutes, and continues like this for 3 hours. If the rungs are each 1 foot apart, how many rungs are under water when the tide stops rising?
Answer: No rungs are under water. The ladder is attached to a boat which rises with the tide, therefore, the bottom rung will stay just above the surface of the water.
Question: You have a cup of tea (A) and a cup of coffee (B) (equal amounts), and you take a spoonful of the coffee (B) and mix it thoroughly into the tea (A). Then you take a spoonful of tea (A) and mix it with the coffee (B). Does the cup that originally had tea in it (A) have more coffee or does the cup that originally had coffee in it (B) have more tea?
Answer: Both cups end up having the same amount of the other liquid in it. When you take the spoonful of coffee (B) and put it in the tea (A) it increases the volume of the tea (A) by that spoonful. Then when you take a spoonful of the tea (A), part of that spoonful is coffee, taking away the proportional amount of coffee. This makes the amount of coffee in the tea and tea in the coffee equal.
John has some chickens that have been laying him plenty of eggs. He wants to give away his eggs to several of his friends, but he wants to give them all the same number of eggs. He figures out that he needs to give 7 of his friends eggs for them to get the same amount, otherwise there is 1 extra egg left.
What is the least number of eggs he needs for this to be true?
Answer: 301 eggs. The number of eggs must be one more than a number that is divisible by 2, 3, 4, 5, and 6 since each of these numbers leave a remainder of 1. For this to be true one less than the number must be divisible by 5, 4, and 3 (6 is 2*3 and 2 is a factor of 4 so they will automatically be a factor). 5 * 4 * 3 = 60. Then you just must find a multiple of 60 such that 60 * n + 1 is divisible by 7. 61 / 7, 121 / 7, 181 / 7, 241 / 7 all leave remainders but 301 / 7 doesn't.
You have a large number of friends coming over and they all get thirsty. Your first friend asks for 1/2 a cup of water. Your second friend asks for 1/4 a cup of water. Your third friend asks for 1/8 a cup of water, etc.
How many cups of water do you need to serve your friends?
Answer: Just one. If your friends kept asking for water like this forever one cup would be enough.
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