Question 885643: A class has 11 students who are to be assigned seating by lot. What is the probability that the students will be arranged in order from shortest to tallest? (Assume that no two students are the same height.) I need an opinion on which of the two answers is correct : First 1/11! Second : 2/11!
The second answer could be argued to be the better answer as two permutations both represent a monotonic sequence one ascending the other declining. In the absence of any association with the students' heights both could correspond to an ascending spatial arrangement. Let's picture a classroom with 11 chairs in a line reaching from the window to the door, with the teacher at the blackboard. His call to line up according to heights could be answered two ways, either the smallest sits at the door or he sits at the window, each corresponding to one permutation and in effect doubling the probability of getting the monotonic order right.
Reducing the problem to the simplest scenario possible let's put a milk jug and a mustard jar on the table. What is the probability that they will be lined up in increasing order of height ?
Without any doubt the answer must be : 2/2! = 1 The experiment confirms it, since try as you might, you can never put them down in anything but an a monotonic order. If they appear to be otherwise, just walk around the table. That is, the increasing/decreasing designation is not the property of the placement but the property of the method of observation.
However, if we put n distinct numbers in an array and ask the same question, then the answer is p=1/n! as the array by convention is accessed from its first location by us or by a computer program. This first element unless otherwise specified is shown on the left side of the printed page as well. There is a reference point which cuts the choices in half by virtue of providing an anchor.
Answer by Edwin McCravy(20060)   (Show Source):
You can put this solution on YOUR website! A class has 11 students who are to be assigned seating by lot. What is the probability that the students will be arranged in order from shortest to tallest? (Assume that no two students are the same height.) I need an opinion on which of the two answers is correct : First 1/11! Second : 2/11!
The second answer could be argued to be the better answer as two permutations both represent a monotonic sequence one ascending the other declining.
In the absence of any association with the students' heights both could correspond to an ascending spatial arrangement. Let's picture a classroom with 11 chairs in a line reaching from the window to the door, with the teacher at the blackboard. His call to line up according to heights could be answered two ways, either the smallest sits at the door or he sits at the window, each corresponding to one permutation and in effect doubling the probability of getting the monotonic order right.
I agree you can look at it that way. That's because tallest to shortest becomes
shortest to tallest if you walk around to the other side of the line.
Reducing the problem to the simplest scenario possible let's put a milk jug and a mustard jar on the table. What is the probability that they will be lined up in increasing order of height ?
Without any doubt the answer must be : 2/2! = 1 The experiment confirms it, since try as you might, you can never put them down in anything but an a monotonic order. If they appear to be otherwise, just walk around the table. That is, the increasing/decreasing designation is not the property of the placement but the property of the method of observation.Yes, that's the same way to look at it. However, if we put n distinct numbers in an array and ask the same question, then the answer is p=1/n! as the array by convention is accessed from its first location by us or by a computer program. This first element unless otherwise specified is shown on the left side of the printed page as well. There is a reference point which cuts the choices in half by virtue of providing an anchor.
I agree that many word problems in mathematics are ambiguous. For instance:
What is the sum of 8 and 12 divided by 4 plus 6 multiplied by 5?
Edwin
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