There are $N$ children. On a particular special day (Christmas e.g.) they are placed in a line (one dimensional) according to uniform distribution. The start of the row is denoted to be the origin of one dimensional space and the length of the line is $L$. After placing them in the line they receive some money according to their distance from the origin. This means that a child that is $m$ meters away from the origin will receive $K-m$ dollars. Now if a child has received $\geq P$ (which is less than $K$) dollars then he can buy a toy. Further it is assumed that if a child has received more than $P$ dollars then he can give the extra money to a child that has received less than $P$ dollars so that the other child (which receives the money from the donor child) can also buy the toy. In this situation my questions are
What will be the average number of children that can buy the toy on any special day?
Can we have an analytical expression for this answer?
Is the provided information sufficient to solve the problem?
Is there any book or paper that has similar kind of problem where I can look for its solution?
Your help in this regard will be much appreciated.
The number of children who can afford to buy a toy (including those who receive donations) is:
$$\min\left(\bigg\lfloor2\left(\frac{K-P}L\right)N\bigg\rfloor, N\right)$$
If $P<K-\frac 12L$, then every child can afford to buy a toy (including those who receive donations).