Question

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?

Answer 1

Let us take there are a total of 100 low income people, of which 50 people are actually needy

By applying the given conditions in question, we can conclude that

1. As 80% of needy had characteristics, we can say that 40 people (80% of 50 people) out of 50 needy people has characteristics. That means 10 needy people will not have characteristics.

2. From above we can also say that 10 people (20% of people) who have characteristics are not needy but belong to low income.

Summarise the above discussion as below

1. Needy with characteristics - 40%

2. Needy without characteristics - 10%

3. Low income with characteristics - 10%

4. Low income & no characteristics - 40%

Off the above four categories, money will reach to 1st and 3rd category (because they are with characteristics)

Under New program, money reached as below

1. Needy with characteristics - 80% of money

2. Low income with characteristics - 20% of money

Under Old program, money reached as below

1. Needy with characteristics - 40% of money

2. Needy without characteristics - 10% of money

3. Low income with characteristics - 10% of money

4. Low income with characteristics - 40% of money

So under new program, needy people received 80% of money and under old program needy people received 50% of money