New Problems

Here are some nice problems I’ve run across recently. They’re from word of mouth, the internet, and books. Only one of them is a “trick” problem. The problems are pretty much unrelated.

Answers here

Do these first two as fast as you can:

  • 999 * 1001
  • A family has five children. What’s the probability that at least three of them are girls?

You have two glass balls and are in a 100-story building. There’s a window on each floor. When you drop the ball out a window, it may or may not break, depending on how high you are. There’s a certain floor that is the transition from not breaking to breaking, so that if you drop it from that floor or above, it will break, and below that floor, it will not. This floor could be any of the floors from one to one hundred with equal probability. What is the fewest number of drops you need to make to be sure you accurately locate the transition floor?

Your friend puts two balls in a jar. Each ball is either red or green, and your friend chooses the color of each ball with a fair coin flip before putting it in. You come up, open the jar, and without looking can smell that there’s at least one red ball in it (but two red balls smell the same as one). What’s the probability that both balls are red?

Same as last problem, but this time you reach in and pull out a red ball. What’s the probability that the remaining ball is also red?

Suppose that there are two barrels, each containing a number of plastic eggs. In both barrels, some eggs are painted blue and the rest are painted red. In the first barrel, 90% of the eggs contain pearls and 20% of the pearl eggs are painted blue. In the second barrel, 45% of the eggs contain pearls and 60% of the empty eggs are painted red. Would you rather have a blue pearl egg from the first or second barrel?

100 prisoners are to be executed, but they are given a chance to save themselves by playing a game. They will all stand in a single file line, so the prisoner in back can see all the other prisoners and the prisoner in front can see no one. The warden will then put a white or black hat on each prisoner’s head, choosing at random as he gets to the prisoner. Then the warden goes to the prisoner at the back of the line (who can see everyone else’s hat, but not his own) and asks him what color his hat is. He can respond only with either “white” or “black”. If he gets it right, he lives. This continues down the line. Each prisoner can hear the responses of all the prisoners who come before him. If the prisoners are allowed to get together and discuss strategy beforehand, how many of the 100 can be saved?

There are eight pitchers of wine, one of which is poisoned. You have some lab rats to test the wine on. If a rat drinks any poison wine, it will die some time within the next 24 hours. How many rats do you need to use to design a test that is certain to discover the poisoned bottle in 24 hours?

Prove that there exist numbers x and y that are both irrational, but x^y is rational.

Suppose you cut a cone out of a sheet of paper. How does the time it takes the cone to fall to the floor when dropped from the ceiling depend on the radius of the cone?

Take a 6*6 chessboard and and 8*8 chessboard. For each, you’re allowed to make one cut through it along the lines between the squares. This will give you four pieces total. How can you make the cuts so those 4 pieces can be rearranged into a 10*10 chessboard? Try the same thing with other Pythagorean triples.

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7 Responses to “New Problems”

  1. Shelley Says:

    While you can use that trick we talked about, 999 x 1001 is made that much easier because 999000 + 999 is really easy.

    Probability of at least three girls is the same as probability of at least three boys: 1/2

  2. Mark Eichenlaub Says:

    like it.

  3. Shelley Says:

    Should have phrased that second one better: if the family doesn’t have at least three girls, it will have at least three boys, and each scenario is equally likely.

  4. Tevong Says:

    I misread that second one as the probability of having 3 girls out of 5 children and promptly started computing the total number of possibilities of girl-boy combinations (2^5) and the total number of ways of choosing 3 girls out of 5 kids (5!/3!), with the ratio of the latter to the former as the answer.

    (I think it’s 5/8th if that makes any sense)

  5. Mark Eichenlaub Says:

    Not quite. The probability of having 3 girls out of 5 kids is actually 5/16. Your error was that 5C3 is actually 5!/(3!*2!). The quick way to answer the question is the one Shelley identified above, though.

  6. Tevong Says:

    ah thanks just came back to correct my error as I had this niggling feeling I made a mistake not thinking through the nCr formula properly..

  7. Answers to Problems « Arcsecond Says:

    […] Answers to Problems By Mark Eichenlaub Answers to these problems […]

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