Knights of the Round Table
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
King Arthur, Merlin, Sir Lancelot, Sir Gawain, and Guinevere decide to go to their favorite restaurant to share some mead and grilled meats. They sit down at a round table for five, and as soon as they do, Lancelot notes, "We sat down around the table in age order! What are the odds of that?"
Merlin smiles broadly. "This is easily solved without any magic." He then shared the answer. What did he say the odds were?
HintDoes it matter if they are sitting clockwise or counterclockwise? Or where the oldest sits?
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Answer
The odds are 11:1. (The probability is 1/12.)
Imagine they sat down in age order, with each person randomly picking a seat. The first person is guaranteed to pick a seat that "works". The second oldest can sit to his right or left, since these five can sit either clockwise or counterclockwise. The probability of picking a seat that works is thus 2/4, or 1/2. The third oldest now has three chairs to choose from, one of which continues the progression in the order determined by the second person, for a probability of 1/3. This leaves two seats for the fourth oldest, or a 1/2 chance. The youngest would thus be guaranteed to sit in the right seat, since there is only one seat left. This gives 1 * 1/2 * 1/3 * 1/2 * 1 = 1/12, or 11:1 odds against.
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Comments
I_am_the_Omega
Dec 13, 2004
 Actually, the odds are exactly 0. Women weren't allowed at the round table. Ever. And, the men who were, always had fixed positions. 
tsimkin
Dec 13, 2004
 0mega  your comments are right if this were the round table of Arthurian meetings, but wrong for the nice little booth in the back of their favorite restaurant. 
monkey41
Dec 17, 2004
 i suck at fractions so ur answer confuzzled me 
I_am_the_Omega
Dec 19, 2004
 ...DOH! I didn't even read the restaurant part :'(
Lol 
cnmne
Dec 28, 2004
 Another approach is to know that there are 5!, or 120, seating arrangements. Since they are sitting at a round table (not the Round Table), there are 5 possible starting points, or 120 / 5, or 24 arrangements that are in the proper order. Since the seating can be in either direction, divide by 2 to get 12. 
nunya
Jan 22, 2005
 is omega(Z)always that much of an egotist?you didnt say THE round table.lighten up!if u think quinevere didnt have anything to do with the round table,think again! 
Takua_Nui
Jan 25, 2005
 I had forgotten to divide by five and two after 5! 
brianz
Apr 30, 2005
 I got it!
Good teaser. 
Cheerleader_09
May 23, 2005
 I got confused because probability and odds aren't the same thing, but it was clever and complicated just the same. 
(user deleted)
May 31, 2005
 I never knew how to work odds out, so i had no idea. But good teaser. 
lessthanjake789
Jun 15, 2005
 maybe i missed something, but there doesnt seem to be a HALF chance of the second oldest, but a FOURTH chance. correct me if im wrong, but couldnt the second oldest sit whereever he please, as in one of the 4 remaining seats? if there was some nuance of the wording that implied he sat HAD to sit next to the oldest, then i guess i'm wrong. also, it says that sat around in age order, but that doesn't so much mean the oldest sat first... 
lessthanjake789
Jun 15, 2005
 maybe i missed something, but there doesnt seem to be a HALF chance of the second oldest, but a FOURTH chance. correct me if im wrong, but couldnt the second oldest sit whereever he please, as in one of the 4 remaining seats? if there was some nuance of the wording that implied he sat HAD to sit next to the oldest, then i guess i'm wrong. also, it says that sat around in age order, but that doesn't so much mean the oldest sat first... 
wordsrcool
Jun 15, 2005
 It's just like the cheap and nasty Mp3 player except it is a round table, (multiply 1/120 by 5) and it can be in clockwise or counterclockwise order. (multiply 5/120 by 2) 
gnosys
Oct 01, 2005
 I'd say the odds are zilch, because Guinevere will inevitably sit between Arthur her husband and Launcelot her lover, both of whom are older than she. Also, would the fact that Merlin  at least according to T.H. White's "The Once and Future King"  is living backwards in time have any effect here? How do you calculate the age of someone who is growing younger  do you still count from his birth, even though that's in the future? 
mr_brainiac
Jan 04, 2006
 Man o man! I got it right on the head , right down to the exact same explanation! Am I briiliant or what? 
Jimbo
Mar 05, 2006
 For those that liek formulas there are (n1)! arragements in a circle = 4! or 24 in this case. One of these orders is from youngest to oldest while another one would be oldest to youngest. Hence probability = 2/24 = 1 in 12. Fantastic puzzle  thanks! 
smoothiemixer11
Jun 12, 2007
 I agree with lessthanjake789. This is wrong because the ratio is not 1/2 for the 2nd oldest  you never said that he had to sit next to the oldest. It should be 1/4 for him, and 1/3 for the third oldest, etc. Please correct this. If I am wrong because you forgot to include something in your question, please tell me.
And Omega (the first comment) it might help if you bothered to read the question before insulting the author like that. 
FerretGirl
Jul 24, 2007
 Tsimkin's reasoning for the probability fraction is correct assuming that everyone sat down in an acceptable manner. What if the middle person sat down second next to the eldest? Or if the youngest sits two seats away? 
javaguru
Dec 08, 2008
 Another way to state the solution is that there are 5! = 120 ways the five can take their seats. The youngest (or oldest) can choose one of five seats and there are two ways to seat the remaining four people. That gives (2x5)/120 = 1/12, or 11:1. Nice puzzle though. 
opqpop
Jan 27, 2010
 Good puzzle. I forgot about clockwise and counterclockwise so ended up with 5/5! at first.
I like the 2/4! solution the best. The (1)(1/2)(1/3)(1/2)(1) solution is not bad either. 
hienvu
Jul 03, 2014
 There is an easier way:
if there are 3 => the chance =1
then the forth comes => he needs to sit at his specific place => chance =1/3
then the fifth comes => he needs to sit at his specific place => chance=1/4
=> total chance =1/12 
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