Multi-set intersection

Question

I have a population of (let’s say, 100 people) with the following three properties:

• Gender: 20 Males & 80 Not-Males
• Ethnicity: 30 Europeans & 70 Non-Europeans
• Height: 40 Tall & 60 Not-Tall

I also have the number of people in groups of two properties in the following table:

I would like to know how to get the number of people in groups of three properties:

    1. Male      &  European      &  Tall
    2. Male      &  European      &  Not-Tall
    3. Male      &  Non-European  &  Tall
    4. Male      &  Non-European  &  Not-Tall
    5. Not-Male  &  European      &  Tall
    6. Not-Male  &  European      &  Not-Tall
    7. Not-Male  &  Non-European  &  Tall
    8. Not-Male  &  Non-European  &  Not-Tall

I have tried to come up with equations to solve for the 8 unknowns but they all cancel together and I get 0=0, 5=5, ... . I am completely mind-blocked!

Answer 1

There‘s not enough information in your table for pairs of properties to deduce the counts of people with triples of properties. To see this, imagine a table with the counts for triples of properties and add $\pm c$ to each entry, with the sign given by the parity of the number of “non”/“not” in the properties. For instance, $c$ is added to the entries for not-male non-European tall people and for male European tall people, but subtracted from the entries for not-male European tall people and for not-male non-European not-tall people. This changes the triple counts but leaves the pair counts unchanged (since each of them is a sum of two triple counts with opposite parity).