Question

A party of $23$ people takes their seats at a round table. The odds against two persons sitting together are
1.$10:1$
2.$1:11$
3.$9:10$
4. none of these

Answer 1

In order to find the odds against two persons, firstly we will be considering the total cases possible of people taking their seats and then we will calculate the favourable cases. Then we will be finding the probability of two persons sitting together. Since we have to find the odds against two persons we will be converting the obtained probability into the odd form and that would be our required solution.

Complete step-by-step solution:
Now let us briefly talk about probability and its types. Probability can be defined as a chance of a particular event to occur from a set of events. The range of probability is between $0$ and $1$. There are three types of probability. They are: theoretical probability, experimental probability and axiomatic probability. We can define an event as something that takes place.
Now let us start solving the problem given.
We can say that since $23$ people can take their seats around the table, the possible cases would be $22!$.
Now, favourable cases would be equal to $2\times 21!$
Now let us find the probability of two people sitting together.
$P=\dfrac{2\times 21!}{22!}=\dfrac{2}{22}=\dfrac{1}{11}$
In order to express the probability as the odds against two persons, we will be expressing the obtained probability as follows:
\Rightarrow $$$$\dfrac{1}{1+10}
$\therefore$ The odds against the two people sitting together is $10:1$.

Note: We must note that we must always express the probabilities in its simplest form. In the above problem, we must not forget to convert the obtained probability in its odd form. We must note that the probabilities are always to be calculated with respect to the total number of outcomes.