Probability of transferring a ball from one bucket to other

Question

There are $n$ buckets. Each of these buckets can hold maximum $k$ balls. The probability that a ball is transferred to $m_{th}$ bucket is denoted as $P(m)$. What is the probability that a ball is transferred from $m_{th}$ bucket to $(m+1)_{th}$ bucket given that $m_{th}$ bucket is full and each ball has equal probability to be transferred from one bucket to another. Please explain your answer.

Thanks,

Answer 1

The question is not well-worded but I take it to be asking for the probability of at least one ball being transferred from the $m^{th}$ to the $(m+1)^{th}$ bucket. Let's call this value $p$.

\begin{eqnarray*} p &=& 1 - P(\mbox{no ball transferred from $m^{th}$ to $(m+1)^{th}$ bucket}) \\&=& 1 - (1 - P(m+1))^k \end{eqnarray*}

because the probability of any particular ball not transferring to the $(m+1)^{th}$ bucket is $1 - P(m+1)$ and all transfers are independent of each other (the way I read it), meaning we can multiply that probability $k$ times, since the $m^{th}$ bucket has $k$ balls.

Answer 2

I think it should be $\dfrac 1 k \cdot P(m+1)$ because there are two events here that happen here,

1. Picking up a ball and 2. Putting it in $m+1_{th}$ bucket. Both events are independent and thus $P(A \cap B) = P(A) \cdot P(B)$