SEARCH FOR "PEN"

Answer: The answer might seem a little counter intuitive at first but we'll see... The short answer is that it is in your advantage to exchange. But why? Well initially there was a 1/3 chance that you were holding the envelope with the note in it and a 2/3 chance that the note was on the table. This is still the case after one of the envelopes on the table has been removed, there is still a 1/3 chance that you have the note and a 2/3 chance of it being on the table. If this is confusing then it may help to think that the questioner knows which envelope the $20 note is in, though in practice it doesn't actually matter. The questioner would always be able to demonstrate that the note was not in one of the envelopes on the table regardless of where the note was, so the fact that he was able to do this changes nothing. Consider a different example.... Say there are a 1000 envelopes on the table, 1 with a note inside. You pick 1 envelope, the chance that this has the note in it is clearly 1/1000, where as the chance that it is still on the table is 999/1000. Odds are its on the table. Now the questioner could, assuming he can remember where the note is demonstrate to you that the note is not in 998 of the envelopes on the table. In this case nothing would have happened to change the fact that there is only a 1/1000 chance of you having the note. That is why you exchange. What is the value of the exchange? Simply before the exchange you have 1/3 of $20 and afterwards you will have 2/3 of $20, ie the advantage to you is about $6.66

Answer: 2,2,9

Answer: The train is moving at 40 miles per hour. Imagine that a friend is walking with you. When the train whistle blows, you head away from the train, he heads toward it. When he reaches safety, you will be 6/8 (or 3/4)of the way across the bridge, and the train will have just reached the bridge. For the train to cross 4/4 of the bridge in the time you cross the remaining 1/4, the train must be moving four times your speed.

Answer: A half-dollar, a quarter, four dimes, and four pennies.

Answer: The plane was flying upside down!

Answer: If you chose to go to the clearing, you're right, but the hard part is correctly calculating your odds.  There are two common incorrect ways of solving this problem.  Wrong answer number one:  Assuming there's a roughly equal number of males and females, the probability of any one frog being either sex is one in two, which is 0.5, or 50%.  And since all frogs are independent of each other, the chance of any one of them being female should still be 50% each time you choose.  This logic actually is correct for the tree stump, but not for the clearing.  Wrong answer two:  First, you saw two frogs in the clearing.  Now you've learned that at least one of them is male, but what are the chances that both are?  If the probability of each individual frog being male is 0.5, then multiplying the two together will give you 0.25, which is one in four, or 25%.  So, you have a 75% chance of getting at least one female and receiving the antidote.  So here's the right answer.  Going for the clearing gives you a two in three chance of survival, or about 67%.  If you're wondering how this could possibly be right, it's because of something called conditional probability.  Let's see how it unfolds.  When we first see the two frogs, there are several possible combinations of male and female. If we write out the full list, we have what mathematicians call the sample space, and as we can see, out of the four possible combinations, only one has two males.  So why was the answer of 75% wrong?  Because the croak gives us additional information.  As soon as we know that one of the frogs is male, that tells us there can't be a pair of females, which means we can eliminate that possibility from the sample space, leaving us with three possible combinations.  Of them, one still has two males, giving us our two in three, or 67% chance of getting a female.  This is how conditional probability works.  You start off with a large sample space that includes every possibility.  But every additional piece of information allows you to eliminate possibilities, shrinking the sample space and increasing the probability of getting a particular combination.  The point is that information affects probability.  And conditional probability isn't just the stuff of abstract mathematical games. It pops up in the real world, as well.  Computers and other devices use conditional probability to detect likely errors in the strings of 1's and 0's that all our data consists of.  And in many of our own life decisions, we use information gained from past experience and our surroundings to narrow down our choices to the best options so that maybe next time, we can avoid eating that poisonous mushroom in the first place.

Answer: Before I left, I wound the wall clock. When I returned, the change in time equaled how long it took to go to my friends house and return, plus the time I spent there. But I knew the latter because I looked at my friends watch when I arrived and left. Subtracting the time of the visit from the time I was absent from my house, and dividing by 2, I obtained the time it took me to return home. I added this time to what my friend watch showed when I left, and set the sum on my wall clock.

Answer: A pelican!

Answer: A 50 cent piece and 2 nickels.

Answer: Abraham Lincoln.