On the occasion of his receiving second Nobel prize, Dr. Linus Pauling, the chemist, remarked that, while the chances of any person in the world receiving his first Nobel prize were one in several billion (the population of the world), the chances of receiving the second Nobel prize were one in several hundred (the total number of living people who had received the prize in the past) and that therefore it was less remarkable to receive one's second prize than one's first.
There are 3 ants at 3 corners of a triangle, they randomly start moving towards another corner.. what is the probability that they don't collide.
randomly found w/0 solution
After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that exactly nine letters were inserted in the proper envelopes?
The two above puzzles w/ links should be cited as the following, per the author's website:
A. Bogomolny, Education, Mathematics, Fun, Pauling, probability from Interactive Mathematics Miscellany and Puzzleshttp://www.cut-the-knot.org/pauling.shtml#solution, Accessed 25 October 2007
A. Bogomolny, The Careless Mailing Clerk from Interactive Mathematics Miscellany and Puzzleshttp://www.cut-the-knot.org/Probability/TenLetters.shtml, Accessed 25 October 2007
5 comments:
ok, so those weren't as hard as I first thought when I read them.
maybe I will post some more later, or someone else could post some... ;-)
For the one with the ants, how random are we talking? Random in that next corner the ant will go to is random, but the ant will follow the side of triangle? Or random in that the direction, speed, corner are random and could at any moment change?
my assumption was that the randomness consisted only of the initial direction that they would take and that they would travel at the same speed and only along the edge of the triangle.
if they could move freely in 2D, then (even if we consider them only moving within the bounds of the triangle) we would have to know more about their relative size, etc.
if they could move at variable speeds, the probability becomes uncalculatable (at least in my brain) since there would be infinite numbers of scenarios where they hit and infinite where they don't
good thoughts though
I am assuming everyone figured these out ... But in case you didn't, or were too lazy to click the solution links, here is what I got:
Nobel) Assuming you knew that the judges were going to pick only a previous winner, then yes, the chances of you winning again are much higher than they were the first time ... but the pure chance of an individual winning both of two drawings (and assuming anyone on earth could win), the probability is really 1:x*x where x=the population of the world.
Ants)Since there are three ants and they can go either left or right this problem is esentially a 3 bit number where only 111 and 000 have them not running into eachother, therefore the probability is 2/8 or 1/4
Letters) In order for 9 to be correct, the 10th must also be correct since there is no where else for it to go, but since exactly 9 are right, this can't happen.
Anyone else think about the problems differently or think I am crazy?
i'd have to disagree with the nobel one... the probability of a person winning, sure, 1 in 6 billion. but the probability of a person winning their second one is 1 in the number of previous winners.
even that i guess i'd have to disagree with. of all people in the world, a SAFE bet is to assume that 75% do not do research, and a safer bet is that 75% of the remaining 25% don't have nobel-quality research. so 6 billion * .25 * .25 = 375 million
just semantics, i guess
also, the phrase "randomly moving" implies direction, velocity, reversal, etc... i think they "begin moving toward a randomly selected other-corner" would be more clear
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