1. ## Probability

1. A party of $\displaystyle n$ mens to be seated round a circular table. Find the odds against the event that two particular men sit together.

2. X and Y stand in a ring with 12 other persons. If the arrangement of the 14 persons is at random, find the chance that there are exactly 5 persons between X and Y.

1. A party of $\displaystyle n$ mens to be seated round a circular table. Find the odds against the event that two particular men sit together.

Seat all but one of the particular men around the table, so now there are
n-1 places where the n-th man could be sat, and two of these are next
to the other particular man. So the requred probability is 2/(n-1) (at least
if n>=4, if n<4 the probability is 1)

RonL

3. So the requred probability is $\displaystyle 2/(n-1)$
That is, the odds against the event is $\displaystyle (n-1):2$. Is this right?

That is, the odds against the event is $\displaystyle (n-1):2$. Is this right?
That is not the standard definition of odds used in statistics.

In statistics the odds of an event are defined to be $\displaystyle p/(1-p)$, so the odds of
the event not happening are $\displaystyle (1-p)/p$ .

$\displaystyle \frac{1-2/(n-1)}{2/(n-2)}=\frac{n-3}{2}$

or: $\displaystyle (n-3):2$.

Though I beleive that in some parts of the gamming industry that odds
are defined differently, but UK bookies use them in the sense that statisticians
use them, or at least they did when I was a child.

RonL