mysterygirl wrote:
Hi,
1:
Bunuel - I went through the link on advanced overlapping set, but I fail to understand the basis for the second formula used in the solution ->
A+B+C- (2-grp overlays)*2 - (3-grp overlays)*3
Please let me know how this formula was derived?
Thanks.
For this reply, I'll be referencing Paresh's brilliant answer, because he goes through the effort of breaking down the various components of our three sets.
My advice is don't start off with the formula as a base to understand the question. Key idea here is understanding the question, and parsing the information in the prompt to reach a valid solution. You read the question, you have an idea of what's being given, you've established that it relates to Overlapping Sets. Great, so what do we have?
The question supplies us with these following knowns
Number of Groups: 3
Group 1 (Sweaty Palms: SP): 40% of 300= 4*30=120
Group 2 (Vomiting: V): 30% of 300 = 3*30 = 90
Group 3 (Dizziness: D): 75% of 300 = 7.5*30=225
All people in the study experience atleast 1 of these symptoms, 35% experience
exactly 2 of those symptoms.
So where are those people that fall in 35% of the study? Referring to Paresh's Venn Diagram:
They are sectors
p+q+r, notice that x was not included? Because x represents people who experienced SP, V, and D. The sectors in bold, represent the people who
exactly two symptoms, for clarity
p: Represents people who experienced both SP and V.
q: Represents people who experienced both SP and D
r: Represents people who experienced both D and V.
The unknowns:
1. People who experienced exactly one of these symptoms.
2. People who experienced all three of these symptoms.
You should get into the habit of not mechanically applying the formula, but take a look at the Venn Diagram .
From the first list of unknowns, where is that represented in the Venn Diagram? Well, obviously, it's going to be as follows:
If we only look at SP, this contains more regions than we would want, we want the part of SP the excludes the overlaps.
Here are the following totals for all three groups.
Total SP: A+p+q+x
Total V: B + p + r + x
Total D: C + q + r + x
Now you should be asking yourself, do we want all those sectors? As you probably assumed, the answer is no. Why don't we want all those sectors? Because the sectors of interest are:
Only SP: A
Only V: B
Only D: C
What's left over?
The intersection of SP and V: p + p + x = 2p + x -> P is in both SP and V
The intersection of SP and D: q + q + x = 2q + x -> Q is in both SP and D
The intersection of D and V: r + r + x = 2r + x -> D is in both D and V.
This is what's meant by exactly two groups overlapping, elements common to two sectors and not the third.
It helps to refer to the Venn Diagram, to see exactly how these intersections are derived. It's easy to see those letters and get lost in the learning process, but this is a way to help distill the information, before you intuitively apply the formula (and not mechanically).
Okay now we sum up what we know.
35% of 300 are in p + q + r, so that means means 105 people in the study had exactly two symptoms.
We know that we care to get the people that are in A, B, and C (Refer to Venn Diagram). So using the supplied in formation
Sweaty Palms has 120 people, both from those 120, there are people who fall in p or q, we don't want that.
Vomiting has 90 people, we obviously don't want all 90. Some fall in the Sweaty palms group (p), others fall in the dizziness group (r).
Dizziness has 225 people, I think you can spot the pattern by now. We don't want the entire set of 225, some are in Sweaty Palms (q), others are in Vomiting (r)
The Sectors we do want are in bold:
SP=
A + p+q + x
V=
B + p + r + x
D =
C + q + r + x
It's like we made a full circle, but we'll start plugging in values.
SP +
V +
D =
A + p+q + x +
B + p + r + x +
C + q + r + xWe know know SP=120, V=90, D=225. We want ONLY A from the Sweaty Palms group, ONLY B from the Vomiting Group, ONLY C from the Diziness group.
Referring the color coded equation above, we do some simple algebraic manipulation to get it in this form.
I moved all the sectors of interest as the first three terms on the RHS.
In the form of sectors:
SP + V +D=
A +
B +
C + p+ p + q + q + r + r + x + x + x
In the form of actual numbers
SP + V+ D= 120 + 90+ 225
SP + V+ D = 435
Now combining the representation that's in the form of sectors, with the representation that's in the form of actual numbers, we get
435 =
A +
B +
C + p+ p + q + q + r + r + x + x + x
You will noticed x is counted three times, because it's where all three groups intersect. x contains people who were dizzy, vomiting, and had sweaty palms. Those poor people
Also, p is counted twice, because p happens to fall in both Sweaty Palms and Vomiting. q happens to be the people the people who were dizzy, and had sweaty palms, and finally r are people who were dizzy and vomiting. While you're reading along, just tick these guys off in the Venn Diagram.
Okay, so we get it in the final form, we're almost there.
435=
A +
B +
C + 2p + 2q + 2r + 3x
435 =
A +
B +
C + 2(p+q+r) + 3x
Now we know Total = 300. We know p+q+r = 105, but it looks like we don't know A + B +C (our areas of interest) AND x. But we can solve for x
Total = A + B + C + x + p + q + r
You might be looking at this, and thinking why is that different from the one we just derived, this one
435 =
A +
B +
C + 2(p+q+r) + 3x
Here were started off the SP, V, D, so within those, some of our sectors were counted twice, and three times as discussed above. The one we will work with now, contains all individual sectors, don't think about groups, just think of the areas they occupy.
Again,
54=
A +
B +
C + x + p + q + r
300 =
A +
B +
C + x + 105
A +
B +
C= 300-105-x
A +
B +
C = 195-x
Looks like we're getting somewhere. Back to this formula, were it contains groups that are counted more than once
435 =
A +
B +
C + 2(p+q+r) + 3x
We will sub in the following
p+q + r = 105
A +
B +
C = 195-x
435= 195-x + 2*(105) + 3x
435=195 + 210 + 2x
435 = 405 + 2x
435- 405 = 2x
30 = 2x
x=15
We know
A +
B +
C = 195-x
and we know x=15
so
A +
B +
C=195-15
A +
B +
C=180
Now I hope you can see why those some sectors were counted twice, and others three times. The post you quoted had:
Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3
Group 1: SP; Group 2: V, Group 3: D
2-Group Overlaps: p+q+r
3-group overlaps: x
Taking the totals of of SP and V and D, subtracting the 2-Group Overlaps (p+q+r) two times and subtracting the three groups overlaps three times
leaves us with the unique regions A,B,C
To summarize, A, B, C were our areas of interest within SP, V, and D. To partition them we utilized totals of Sweaty Palms, which contained regions that fell in two other sectors, and one region that was shared by all three sectors. Same from the other Groups of Symptoms.
This should not be your process when solving it, but it helps to break it down while you're studying. So during the actual exam, you'll know how to parse the question, what information to utilize and HOW to utilize it to reach the answer.
close
18 Aug 2013, 05:53
. Approach credits - Genoa2000
