Question

Suppose that three red balls and three white balls are thrown at random into three boxes and and that all throws are independent. What is the probability that each box contains one red ball and one white ball?

Short answer

The probability of each box containing one red ball and one white ball is 0.049383

Step-by-step solution

Given information

Here, three red balls and three white balls are randomly thrown into three boxes.

Thrown are independent of each other.

Finding the probability of each box containing one red ball and one white ball

The probability of each box containing one red ball is,

\(\begin{aligned}{}\Pr \left( {Each\,\,box\,\,contains\,\,one\,\,red\,\,ball} \right) = \frac{{3!}}{{{3^3}}}\\ = \frac{6}{{27}}\\ = \frac{2}{9}\end{aligned}\)

Similarly, the probability of each box containing one white ball is,

\(\begin{aligned}{}\Pr \left( {Each\,\,box\,\,contains\,\,one\,\,white\,\,ball} \right) = \frac{{3!}}{{{3^3}}}\\ = \frac{6}{{27}}\\ = \frac{2}{9}\end{aligned}\)

So, the probability of each box containing one red ball and one white ball is,

\(\begin{aligned}{}\Pr \left( {Each\,\,box\,\,contains\,\,one\,\,red\,\,and\,\,one\,\,white\,\,ball} \right) = \frac{2}{9} \times \frac{2}{9}\\ = \frac{4}{{81}}\\ = 0.049383\end{aligned}\)

Therefore, the probability of each box containing one red ball and one white ball is 0.049383