Imagine that the government is initially transferring a fixed dollar amount to every person who is observed to have a low income. The government believes that everyone who is genuinely needy has a low income, but that only half of low-income people are genuinely needy. It decides to give a transfer only to people who have certain characteristics, such as advanced age and physical disabilities. The distribution of these characteristics has the following properties:

Everyone who has the characteristics is low income.

Half of the low-income people have the characteristics.

Eighty percent of the needy have the characteristics.

The government also doubles the transfer given to each eligible person, so that the total amount of money that it transfers remains the same. The low-income population can be divided into four groups: genuinely needy with the required characteristics, genuinely needy without these characteristics, low-income with the required characteristics, and low income without the required characteristics. What percentage of the low-income group falls into each category? What fraction of the money transferred accrued to the people in each group under the old program, and what fraction accrues under the new program?