In a theater, the setting for Acts II-V of King Lear is determined by a draw before the acts. The auditorium is divided into four sections, and the audience moves to another area with their seats if the section they are sitting in is chosen as the next setting. Let's assume that all four areas are large enough that if we select one, we can move our seat there, and we choose each area with equal probability. How many times more likely is it that we have to "move" twice, compared to only once?
Note. The task was formulated inaccurately. It is not clear from the text whether a location can be drawn multiple times or not. The majority of the contestants solved the task assuming that a location can be drawn multiple times, and the solution below also follows this assumption. There were those who solved the task assuming that a location can only be drawn once before the 2nd to 5th draws.

Solution. The probability that we have to move before a given draw is 1/4; the probability that we do not have to move is $3 / 4$.

The one out of the four draws that will be the one before which we have to move can be chosen in $\binom{4}{1}=4$ ways. Thus,

$$
P_{\text {move once }}=4 \cdot \frac{1}{4} \cdot\left(\frac{3}{4}\right)^{3}=\left(\frac{3}{4}\right)^{3}
$$

Similarly:

$$
P_{\text {move twice }}=\binom{4}{2} \cdot\left(\frac{1}{4}\right)^{2} \cdot\left(\frac{3}{4}\right)^{2}
$$

Therefore,

$$
\frac{P_{\text {move twice }}}{P_{\text {move once }}}=\frac{\binom{4}{2} \cdot\left(\frac{1}{4}\right)^{2} \cdot\left(\frac{3}{4}\right)^{2}}{\left(\frac{3}{4}\right)^{3}}=\frac{\binom{4}{2} \cdot\left(\frac{1}{4}\right)^{2}}{\frac{3}{4}}=\frac{6 \cdot \frac{1}{4^{2}}}{\frac{3}{4}}=\frac{1}{2}
$$

Thus, the probability that we have to move twice is half as large as the probability that we have to move only once.