Here is the 2 part question. Given n people are in a straight line what is the probability
A: They are next to eachother
B: They are seperated by exactly one person
For a i worked it out a few ways and it looks like 2/n but im not sure why and part b im kinda lost on how to figure it out.
If they are in a straight line but not next to each other,
and we only have the value n,
the problem to me is undefined as seperation from each other has
as many possibilities as one wishes.
Maybe i misread the question,
but i'd expect a subgroup to be examined from the entire group.
Sorry i wasnt very specific. Here is the full question. Given 2 people person a and person b are standing in a straight line with n people. What is the probability they are standing next to eachother. And the second part is what is the probability they are seperated by exactly 1 person.
There are n! arrangements of n people.
If 2 of these n are considered as a pair, then they are a "unit" out of n-1.
They then have n-1 possible positions as a unit in the line of n people.
There are 2! arrangements of the 2 people side by side,
hence the first probability is
times the remaining (n-2) people arranged ......added in hindsight
Having 1 person between them....
Take a "unit" as 3.
There are n-2 such "units", with only 2! variations of that "unit" countable.
However, this "unit" can be in n-2 positions.
Hence, the probability is
times the remaining (n-3) people arranged ......added in hindsight
I forgot to arrange the remaining (n-2) and (n-3) people... sorry.
Plato is absolutely correct!
Here is the resoning . . .
There are arrangements of the people.Given people standing in a line.
Duct-tape and together.(a) What is the probability that two particular people, and , are adjacent?
They can be taped like this: . . . . 2 choices.
Then we have "people" to arrange.
. . .There are: . ways.
Hence, there are: . ways for and to be adjacent.
(b) What is the probability they are seperated by exactly one person?
If there is exactly one person between A and B,
. . they can be arranged: . . . . 2 choices.
Then can be any of the other people.
And we have "people" to arrange.
. . There are: . ways.
Hence, there are: . ways for and to be separated by one person.