How to determine number of customers that buy two or more products out of four possible products with limited data points?

Question

I have this tough math problem that I've been thinking on and haven't been able to figure out. Have looked into overlapping sets, venn diagrams, combinations, etc.

Let's say there are 2.207 million total customers who may buy between 1 and 4 of the following products: A, B, C and D.

If the customer counts for each product are:

A (2.021 million) (not customers that only buy A. These are all customers that have A as at least one of their products.)

B (650k)

C (1.286 million)

D (176k)

How can I figure out how what percent of the 2.207 million total customers buy at least two products? Is it possible with this data?

I've tried calculating probability of customers who buy only A, only AB, only DB, etc. with this venn diagram:

I've linked an image 1 that shows my attempt. In the outside main circles, you see the number of total customers that buy each product, the % of total customers that buy each product, and the % that I believe buy only those products. I feel like this approach has to be wrong in some way though.

Thank you in advance!!

Answer 1

This count cannot be determined exactly. It can range from 1.1 million to 1.926 million.

Here's a way to obtain 1.1 million:

A:   921k 
C:   186k 
AC:  274k 
ABC: 650k 
ACD: 176k 

Here's a way to obtain 1.926 million:

A:   271k 
B:    10k 
AB:  640k 
AC: 1110k 
CD:  176k