{Four sample-selected, questions}
 
You are in a boat with a rock, on a fresh water lake. You throw the rock into the lake. With respect to the land {shore}, does the water level increase, decrease, or stay the same? My co-workers think that the water level will stay the same. - David
 
The water level relative to the shore will decrease. Inside the boat the rock is pressing down on the canoe and thus pushing up the water around it. The amount of water displaced is equal in weight to that of the rock. For example, a 10 pound rock will displace 10 pounds of water upward. When the rock is thrown overboard the weight will not matter but rather the volume of the rock. So the rock will push upward an amount of water equal in volume to the rock. The mass of a rock is greater than that of water so the rock displaces more water pushing down on it than in it. So the level of the lake will be higher with the rock in the canoe than at the bottom of the lake. May 10, 2006
 
 
What do you think of the Bible Code? - Vince from Manila
 
I would put those behind it on the same level as those selling get rich quick gambling schemes. The mathematically ignorant taking advantage of the mathematically ignorant. Nov. 22, 2005
 
 
My question is about a problem that is known as the "two envelope paradox". You are on a game show. In front of you are 2 envelopes, each containing an unknown amount of cash. You are told that 1 envelope has twice as much money as the other. You are now asked to choose an envelope. You choose one. It contains $50,000. Now you are told that you can keep the envelope you picked, or swap for the other one. Should you swap? Knowing ahead of time that you could swap, then it doesn't matter, as you would just choose the envelope you ultimately want. But because you only find out about swapping after you choose an envelope, then the original selection and the option to swap are 2 independent events, correct? That said, when deciding to swap or not, the other envelope contains either twice as much or half as much as what you currently have. So it has either $100K or $25k. Since there is a 50% chance of either occurring, the Expected Value of the other envelope is $62,500. Generically speaking, if we let x = the amount you originally selected, then the other envelope's EV is 1.25x. Therefore it is always correct to swap. Is this correct? Thank you. - Derek from Boston
 
 
I'm very familiar with this problem. I address it on my web site of math problems, problem number 6. There I address the general case, including not looking in the first envelope at all. However to answer your question we can not ignore the venue of where the game is taking place. You said it was a "game show." On most game shows $50,000 is a nice win. Few contestants on the Price is Right ever make it that high. I would guess that fewer than 50% of players on Who Wants to be a Millionaire get that high. Meanwhile wins of $25,000 are not unusual on game shows. Cars are won routinely on the Price is Right, which have values of about $25,000. The $32,000 level is a common win on Who Wants to be a Millionaire. The average win on Jeopardy per show is roughly $25,000. The great Ken Jennings averaged only $34,091 over his 74 wins. So, my point is that $50,000 is a nice win for a game show, and $100,000 wins are seen much less often that $25,000. Thus as a game show connoisseur it is my opinion that the other envelope is more likely to have $25,000 than $100,000. So I say in your example it is better to keep the $50,000. It also goes to show you can never assume the chances that the other envelope has half as much or twice as much are exactly 50/50. Once you see the amount and put it in the context of the venue it is being played you can make an intelligent decision on switching, which throws the 1.25x argument out the window. Nov. 2, 2005
 
The radius of the circle is 1. The triangle is equilateral. Find the area of each colored region.
 
I don't want to blow the answer for those who want to solve it for themselves. For the answer and solution visit my other web site mathproblems.info, problem 189. Aug. 28, 2005
 
{once after having clicked on the link below, a further 'site'-search will be necessary.. in order to access these specific questions; so much for my ability to remember long 'URLS' without starting over, and employing the 'tried-and-true' method, of writing it down on some 'scratch' paper!} http://wizardofodds.com/askthewizard/