what is probablity of red ball between 2 white balls

Question

We have 4 Red balls and 4 white balls. what is the probability of the arrangments that have a red ball in the middle of 2 white balls?

Answer 1

Let us count the words that do not contain the substring $z:=WRW$. These words are of the form $$r\ W\ x\ W\ x'\ W\ x''\ W\ r'\ ,$$ where the $r$-substrings contain an arbitrary number of $R$s, whereas the $x$-substrings contain $0$, $2$, $3$, or $4$ $R$s, and there are $4$ $R$s in all.

If all $x$-substrings are empty then $r=R^k$ with $0\leq k\leq 4$; makes $5$ words.

If exactly one $x$-substring is $=R^2$ and the others are empty then $r=R^k$ with $0\leq k\leq 2$; makes $3\cdot 3=9$ words.

If two $x$-substrings are $=R^2$ and the third is empty then the $R$s are used up; makes $3$ words.

If one $x$-substring is $=R^3$ then either $r$ or $r'$ is $=R$; makes $3\cdot2=6$ words.

If one $x$-substring is $=R^4$ then the $R$'s are used up; makes $3$ words.

It follows that there are $26$ words containing $4$ $W$s and $4$ $R$s, but not the substring $z$. Since there are ${8\choose4}=70$ arrangements of the $8$ letters in all the required probability $p$ is given by $$p={70-26\over 70}={22\over35}\ .$$