and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
Y by
Each of 
balls is placed in on of 
bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
balls is placed in on of 
bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
This post has been edited 1 time. Last edited by jlacosta, Nov 15, 2023, 4:41 PM
Reason: Standardizing source field
Reason: Standardizing source field
.
chance of having an odd number of balls, then the third bin must be odd, for a total probability of 
.
.
is around 
be the set of distributions with two bins with nonzero even numbers of balls. Clearly, this is almost all the distributions containing a bin with an even number of balls. Let 
be the set of distributions where all bins have distinct odd numbers of balls, and similarly 
defined by moving a ball from the even bin with a larger number of balls to the other even bin. This maps each element of 
where 
is mapped to by 
.
,
.
ball and adding two balls at a time. There are two possible states: let state 
be when two bins are even and one is odd and let state 
chance of moving to state 
chance of staying at state 
be the desired probability. Because there are ![\[ p = \frac 13 p + \frac 29 (1-p) \Rightarrow p = \boxed{\frac 14}.\]](http://latex.artofproblemsolving.com/6/1/6/6163ac8a5cee2c0ea50b2b068ccdc70a70feb9d8.png)
has 
ways. The desired value is 
which is closest to 
