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- anonymous

A group of 1000 patients each diagnosed with a certain disease is being analyzed with regard to the disease symptoms present. The symptoms are labeled A,B,and C, and each patient has at least one symptom. Also,
900 have either symptom A or B (or both)
900 have either symptom A or C (or both)
800 have either symptom B or C (or both)
650 have symptom A
500 have symptom B
550 have symptom C
Determine:
1. The number who had both symptoms A and B
2. The number who had either symptom A or B (or both) but not C
3. The number who had all three symptoms
I cant seem to figure this out..

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- anonymous

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- anonymous

1. (just A + just B)-(900 have either symptom A or B (or both)= 250

- anonymous

what does your number 2 mean?

- anonymous

|dw:1328113317366:dw|

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- anonymous

if you continue this pattern tho, and 500 have B, your already at 500 right? so that leavesno one having all three or just B? That is why I am confused. PLEASE HELP :)

- anonymous

i'm thinking............

- mathmate

You can apply the principle of inclusion/exclusion.
For 1,
\(|A \cup B| = |A|+ |B| - |A \cap B| \) =>
\(|A \cap B| = |A|+ |B| - |A \cup B| \) =650+500-900=250
Similarly
\(|B \cap C| = |B|+ |C| - |B \cup C| \) = 500+550-900=150
and
\(|C \cap A| = |C|+ |A| - |C \cup A| \) = 550+650-800=400
For 2,
The answer is in the given data.
For 3,
\(|A \cup B \cup C| = |A|+ |B| + |C| - |A \cap B|- |B \cap C|- |C \cap A| + |A\cap B \cap C| \) which means
\(|A \cap B \cap C| = |A \cup B \cup C| - |A|- |B| - |C| + |A \cap B|+ |B \cap C|+ |C \cap A| \)
=1000-650-500-550+250+150+400 = 100
For more information, see
http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle

- anonymous

omg. i didn't mentioned that all of the patients are 1000

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