Analyzing Customer Product Purchases Using Venn Diagrams and Basic Probability

[InPhase]

Homework Statement

A salesperson with a three-product line calls on 200 customers over a period of time. These 200 customers place orders as follows:

100 ordered product A
95 ordered product B
85 ordered product C
50 ordered products A and B
55 ordered products A and C
30 ordered products B and C
20 ordered products A, B, and C

Determine the number of customers who order: (a) at least one product, (b) no products, (c) exactly one product, (d) exactly two products, (e) exactly three products.

Use a venn diagram to illustrate...

The Attempt at a Solution

I drew a venn diagram showing the overlaps in the product purchases. I also took the ratios between the number of products to the customers, e.g. 100/200, 95/200, 85/200 etc. But I have no idea how to apply these to get the answers. I just guessed that those ratios would be significant somehow. This comes from a worksheet, but the textbook is only covering set theory. We haven't discussed this in class either, so I'm stuck.

 

[Ibix]

I think you're over-complicating it. If you've managed to draw the Venn diagram, you've done everything you need. The question is only asking for numbers of people who bought x products, so all you need to do is work out which areas of the Venn diagram correspond to "ordered one or more products" (etc) and add up the numbers in those areas.

Sometimes shading the diagram is helpful to keep track of your thinking. Draw a copy for each question and scribble out (careful not to go outside the lines!) areas that don't contribute to the number you are currently calculating.

 

[statdad]

Or begin labeling the Venn Diagram with counts for each group: start with the most restrictive information (all three) first.

 

[InPhase]

Thank you both for the response! I have labeled the diagram according to the information given in the problem, however, I am stuck when it comes to coaxing out the answers to the questions. For example, 20 customers ordered all three products, but since the total number of orders exceeds the number of total customers, I can't figure out how to determine how many ordered exactly one, or zero, products etc. It seems to me there is information missing from the problem, but that is just guessing.

 

[Ibix]

Read the question carefully. What it does not say is just as important as what it does.

For example, it says 30 people ordered B and C. It does not say that 30 people ordered B and C and not A. Can you figure out how many people ordered B and C and not A?