In a random Sample of Canadians
30% have Beeburr CD's (B)
35% have Fergie CD's (F)
55% have Motorhead CD's (M)
5% have B and F
10% have B and M
10% have F and M
95% have at least one artist
place in Venn Diagram Find Number and Describe:
a) _____
n(BuFuM)
b)Exactly 2 of 3 artists (Find # only)
To answer these questions, we need to use the principles of set theory and Venn diagrams. Here's how you can find the answers:
a) To find the number and describe the region for n(BuFuM), which represents the number of Canadians who have all three artists' CDs, we need to determine the overlapping area of the three sets B, F, and M.
1. Start by drawing a Venn diagram with three overlapping circles representing B, F, and M. Label these circles accordingly.
2. Based on the given percentages, we know that 30% have B, 35% have F, and 55% have M.
3. To find the overlapping region, identify the percentages of the intersections between the circles. Let's calculate these:
- From the given information, we know that 5% have both B and F, so the intersection of B and F is 5%.
- Similarly, 10% have both B and M, so the intersection of B and M is 10%.
- Also, 10% have both F and M, so the intersection of F and M is 10%.
- Finally, we know that 95% have at least one artist, which means that 100% - 95% = 5% have none of the three artists.
4. Now, let's calculate the overlapping region n(BuFuM):
- Start with the total percentage of the region covered by B, F, and M, which is 30% + 35% + 55% = 120% (including overlaps).
- Then, subtract the individual overlap percentages: 120% - 5% (B and F) - 10% (B and M) - 10% (F and M) = 120% - 25% = 95%.
Therefore, n(BuFuM) is 95%.
b) To find the number of Canadians who have exactly two out of the three artists' CDs, we need to add up the percentages of the overlapping regions between two circles:
1. Identify the overlaps between two sets:
- From the given information, we already know the percentages of the intersections between B and F, B and M, and F and M, which are 5%, 10%, and 10%, respectively.
2. Add up these overlap percentages: 5% + 10% + 10% = 25%.
Therefore, the number of Canadians who have exactly two out of the three artists' CDs is 25%.