Question

plications of probability and geometry
the venn diagram shows the number of patients seen at a pediatrician’s office in one week for colds, c, ear infections, e, and allergies, a.
how many patients had allergies or ear infections, but not both?

Explanation:

Step1: Identify regions for allergies only and ear infections only

For allergies (A) only: The region is 15 (A only) + 2 (A and C only) + 3 (A and E only)? Wait, no, wait. Wait, the question is allergies or ear infections but not both. So we need patients with allergies only, ear infections only, and also the parts of allergies that don't overlap with ear infections, and ear infections that don't overlap with allergies. Wait, let's look at the Venn diagram. The circles are C (colds), E (ear infections), A (allergies).

First, find the number of patients with allergies (A) but not ear infections (E): That would be the parts of A not overlapping with E. So A's regions: 15 (only A), 2 (A and C only), 1 (A, C, E)? Wait no, the overlapping regions: the middle is 1 (all three), then A and C only is 2, A and E only is 3, E only is 9, C only is 4, C and E only is 10, A only is 15, and outside is 6. Wait, no, let's parse the Venn:

- C (colds) circle: 4 (only C), 10 (C and E only), 2 (C and A only), 1 (all three)
- E (ear infections) circle: 9 (only E), 10 (C and E only), 3 (E and A only), 1 (all three)
- A (allergies) circle: 15 (only A), 2 (C and A only), 3 (E and A only), 1 (all three)
- Outside: 6

Now, the question is "allergies or ear infections, but not both". So that means (allergies only) + (ear infections only) + (allergies and colds only) + (ear infections and colds only)? Wait no, "or" in logic is inclusive, but "but not both" means exclusive or: (allergies only) + (ear infections only) + (allergies and colds only) + (ear infections and colds only)? Wait, no. Let's think: "allergies or ear infections, but not both" means (allergies and not ear infections) + (ear infections and not allergies).

So first, allergies and not ear infections: parts of A not in E. So A's regions not in E: only A (15) + C and A only (2) = 15 + 2 = 17.

Then, ear infections and not allergies: parts of E not in A. So E's regions not in A: only E (9) + C and E only (10) = 9 + 10 = 19.

Wait, no, wait: A and not E: A \ E = (A only) + (A and C only) = 15 + 2 = 17.

E and not A: E \ A = (E only) + (E and C only) = 9 + 10 = 19.

Then sum these two: 17 + 19 = 36? Wait, but let's check again. Wait, maybe I misread the Venn. Wait the Venn diagram:

Looking at the image:

- C (blue) has 4 (only C), 10 (C and E), 2 (C and A), 1 (all three)
- E (red) has 9 (only E), 10 (C and E), 3 (E and A), 1 (all three)
- A (green) has 15 (only A), 2 (C and A), 3 (E and A), 1 (all three)
- Outside: 6

So allergies and not ear infections: A \ E = (only A) + (C and A only) = 15 + 2 = 17.

Ear infections and not allergies: E \ A = (only E) + (C and E only) = 9 + 10 = 19.

Now sum these two: 17 + 19 = 36. Let's check the options: 36 is one of the options. So that's the answer.

Wait, let's verify:

Allergies or ear infections, but not both: (A ∪ E) - (A ∩ E). The formula for exclusive or is |A| + |E| - 2|A ∩ E|. Let's compute |A|: only A (15) + C and A (2) + E and A (3) + all three (1) = 15+2+3+1=21. |E|: only E (9) + C and E (10) + E and A (3) + all three (1) = 9+10+3+1=23. |A ∩ E|: E and A (3) + all three (1) = 4. So |A| + |E| - 2|A ∩ E| = 21 + 23 - 2*4 = 44 - 8 = 36. Yes, that matches. So the answer is 36.

Step1: Calculate |A| (allergies)

|A| = only A (15) + C&A (2) + E&A (3) + all three (1) = 15 + 2 + 3 + 1 = 21.

Step2: Calculate |E| (ear infections)

|E| = only E (9) + C&E (10) + E&A (3) + all three (1) = 9 + 10 + 3 + 1 = 23.

Step3: Calculate |A ∩ E| (both allergies and ear infections)

|A ∩ E| = E&A (3) + all three (1) = 3 + 1 = 4.

Step4: Calculate exclusive or (allergies or ear infections, not both)

Using formula: |A| + |E| - 2|A ∩ E| = 21 + 23 - 2*4 = 44 - 8 = 36.

Answer:

36