Q. 2.43 - A First Course in Probability

Question

$\left(a\right)$ If $N$people, including $A$and$B$, are randomly arranged in a line, what is the probability that $A$and $B$are next to each other?

$\left(b\right)$ What would the probability be if the people were randomly arranged in a circle?

Step-by-Step Solution

Answer

$\left(a\right)$the probability that $A$and $B$are next to each other is$\frac{2}{N}$.

$\left(b\right)$the probability is if the people were randomly arranged in a circle is$\frac{2}{N-1}$.

Step 1 Given Information.

$N$ people, including $A$and$B$, are randomly arranged in a line.

Step 2 Part (a) Explanation.

$$\begin{gathered} P(A and B sitting together)=\left[2*(N-1)!\right]/(N!) \\ =2/N \end{gathered}$$

Step 3 Part (b) Explanation.

$$\begin{gathered} P (A and B sitting together in a circle)=\left[2*(N-2)!\right]/\left[(N-1)!\right] \\ =2/(N-1) \end{gathered}$$