  Consider an ephemeral sale-and-repurchase of a security resulting in the same
position before the sale and after the repurchase. A sale-and-repurchase is a
wash sale if these transactions result in a loss within $\pm 30$ calendar days.
Since a portfolio is essentially the same after a wash sale, any tax advantage
from such a loss is not allowed. That is, after a wash sale a portfolio is
unchanged so any loss captured by the wash sale is deemed to be solely for tax
advantage and not investment purposes.
  This paper starts by exploring variations of the birthday problem to model
wash sales. The birthday problem is: Determine the number of independent and
identically distributed random variables required so there is a probability of
at least 1/2 that two or more of these random variables share the same outcome.
This paper gives necessary conditions for wash sales based on variations on the
birthday problem. This allows us to answer questions such as: What is the
likelihood of a wash sale in an unmanaged portfolio where purchases and sales
are independent, uniform, and random? This paper ends by exploring the
Littlewood-Offord problem as it relates capital gains and losses with wash
sales.
