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I have another explanation for this.

Assume out of n persons the person at position 1 is selected. P of doign this = 1/n Then we can select anyone other than the person sitting next to him. P of doing this = (n-2)/(n-1). The same is true for the person who is last in the row.
Net P = 2 * 1/n * (n-2)/(n-1)

Now we pick a person from the middle of the row. P of doing this is (n-2)/n
Then se can select any one who is not on either side of this person. P of doing this = (n-3)/(n-1)
Net P = (n-2)/n * (n-3)/(n-1)

Total P = 2 * 1/n * (n-2)/(n-1) + (n-2)/n * (n-3)/(n-1)

= (n-2)/( n * (n-1) ) [ 2 + n-3 ]
= (n-2)/n = 1-2/n



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