Combination Practice Question (Venn Diagram Help)

Question

1.A survey of 1000 television viewers conducted by a local television station produced the following data:

• 40% watch the news at 12:00

• 60% watch the news at 18:00

• 50% watch the news at 23:00

• 25% watch the news at 12:00 and at 18:00

• 20%watch the news at 12:00 and 23:00

• 20% watch the news at 18:00 and 23:00

• 10% watch all three news broadcasts

I'm just so confused on how to create a Venn diagram with the following percentages/data,

I know you have to solve for the eight groups, but I find it hard to understand, like for example AB'C' = A - A(B ∪ C) = A - AB - AC + ABC = 40% - 25% - 20% + 10% = 5%, etc, I'm confused on the ABC' or AB'C', etc, I don't get what it means and why you have to do that. Please if someone can explain, that would be great.

Answer 1

So I find the easiest way to start by placing your 10% who watched all 3 times in the center, the intersection of all 3 circles.

You can then work out the places where 2 circles intersect, such as how many watched at 12:00 and 18:00, which totals 25%. So if 10% are already counted (from watching all 3 times) that leaves 15%. This means 15% of people only watched at 12:00 and 18:00.The same applies to 18:00 and 23:00, 20% total. 10% are already in the center so only 10% remain who only watched at 18:00 and 23:00. Finally, 12:00 and 23:00 is 20%, again, this leaves 10% who only watched at 12:00 and 23:00.

You can then fill in the groups who only watched a single broadcast, aka have no overlaps in the venn diagram. So for 12:00 that was 40% total. 10% watched all 3 times, 15% watched at 12:00 and 18:00, and 10% watched at 12:00 and 23:00. That leaves you with 5% who only watched the 12 time.

18:00 is 60% total so 60%-10%-15%-10% = 25% only watched at 18:00.

23:00 is 50% total so 50%-10%-10%-10% = 20% only watched at 23:00.