probability

[JeffM]

Take things step by step.

We select an arbitrary pair, A and B.

There are two mutually exclusive cases: A and B are in the same line or different lines.

What are the probabilities?

Whatever position and line A is in, B has four ways to be in the same line and five ways to be in the other line. So the probability of same line is 4/9 and the probability of different line is 5/9.

Same line.

The number of ways to pick two positions from five is 10. (1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3;5; 4,5)

How many of those are adjacent. Just four. Probability of being adjacent given in same line = 4/10 = 2/5.

(4/9) * (2/5) = 8/45.

Different lines.

How many ways to pick a position in one line? Five. How many ways to pick a position in other line? Five. Twenty-five possibilities, only five of which are the same in both lines. Probability of being adjacent given in different lines = 5/25 = 1/5.

(5/9) * (1/5) = 5/45.

(8/45) + (5/45) = 13/45.

 

[hdadashi]

2!= that two random guys can change their places in 2! situations.
Combination(8,1)= we choose 1 place out of 8 possible places that they can stand.
Combination(5,1)= we choose 1 place out of 5 possible places that they can stand.
8!= other 8 guys can change their places in 8! situations.
10!= all 10 guys can change their places in 10! situations.

 

[Dr.Peterson]

I think the answer is as follow:
P1=Probability of being next to each other=View attachment 15666=8/45
P2=Probability of being in front of each other=View attachment 15667=5/45
total answer=P1+P2=View attachment 15668

2!= that two random guys can change their places in 2! situations.
Combination(8,1)= we choose 1 place out of 8 possible places that they can stand.
Combination(5,1)= we choose 1 place out of 5 possible places that they can stand.
8!= other 8 guys can change their places in 8! situations.
10!= all 10 guys can change their places in 10! situations.

It is not clear what "they" refers to! There are certainly more than 8 places where two people can stand, or even one person (if you are using the singular "they").

It can be difficult, but very important, to state your reasoning in a convincing way, so you need to learn to do so.

Do you agree with me that the 8/45 should be the probability of being in front of each other, and the 5/45 should be the probability of being next to each other?

As I interpret what you have said here, I think you are using permutations, so that the denominator is the number of ways to arrange all 10 people, while the numerator is the number of ways to place everyone so that the two people are next to one another. But I don't see it yet; I understand 8! ways to place the other 8 after placing the two, but how are there only 2!*8 places for the two?

For the numerator, I would have said that there are 10 choices for the first person, 1 for the second (next to him), and 8! for the rest.