Question

Suppose that a certain boxAcontains five balls and another boxBcontains 10 balls. One of these two boxes isselected at random and one ball from the selected box is transferred to the other box. If this process of selecting abox at random and transferring one ball from that box to the other box is repeated indefinitely, what is the probability that boxAwill become empty before boxBbecomes empty?

Short answer

The probability that before box B will become empty, box A will become empty is \(\frac{2}{3}\).

Step-by-step solution

Given information

There are 5 balls in box A and 10 balls in box B.

When box A is selected then one ball from box A is transferred to box B. Similarly the same for box B.

So, here\(i = 5\),\(k = \left( {5 + 10} \right) = 15\) and \(p = \frac{1}{2}\).

Compute the probability

Referring to equation 2.4.6 for the equation which is given below,

\({a_i} = \frac{i}{k};i = 1,2,...,k - 1\)

Here,\(a = \frac{5}{{15}} = \frac{1}{3}\)

So, the probability that box B will become empty is\(\frac{1}{3}\).

The probability that at first box A will become empty is

\(\begin{aligned}{}1 - a = 1 - \frac{1}{3}\\ = \frac{2}{3}\end{aligned}\)

Therefore, the probability that before box B will become empty, box A will become empty is \(\frac{2}{3}\).