1 The following table summarizes visits to emergency departments at four hospita
ID: 2959651 • Letter: 1
Question
1 The following table summarizes visits to emergency departments at four hospitals in Arizona. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital. [5 points]
1 2 3 4 Total
Total 5292 6991 5640 4329 22,252
LWBS 195 270 246 242 953
Admitted 1277 1558 666 984 4485
Not admitted 3820 5163 4728 3103 16,814
Let A denote the event that a visit is to Hospital 1 and Let B denote the event that a visit results in admittance to any hospital. Determine the number of persons in each of the following events.
(a) A n B
(b) A'
(c) A u B
(d) A u B'
(e) A' n B'
.
Explanation / Answer
Parts (c) and (d) are missing the logical operator -- I think maybe it's a special character that my computer can't read. I'm going to assume it's the "OR" or union symbol. If that's not the case let me know and I'll edit my answer.
The set A is people visiting Hospital 1 (first column) and set B is people who are admitted to a hospital (third row).
a) A n B = the people who visit Hospital 1 AND are admitted to a hospital. Therefore they must be the people admitted to hospital 1, or 1277 people.
b) A' = the complement of A, or the people in the sample space who DON'T visit Hospital 1.
People who don't visit Hospital 1 = Total people - People who visit Hospital 1
= 22,252 - 5292
= 16960.
c) A U B = the people who visit Hospital 1 OR are admitted to a hospital.
People who visit Hospital 1 or are admitted anywhere = Total people who visit Hospital 1 + Total people who are admitted - (People who both visit Hospital 1 and are admitted)
We have to subtract the last quanitty as otherwise we are counting those people twice, once for visiting Hospital 1 and once for being admitted.
People who visit Hospital 1 or are admitted anywhere = 5292 + 4485 - 1277
= 8500.
d) A U B' = the people who visit Hospital 1 OR the people who are NOT admitted to any hospital.
Total = Number of people who visit Hospital 1 + Number of people not admitted anywhere - (Number of people who both visit Hospital 1 and are not admitted)
As addition is commutative, we can rewrite this as,
Total = (Number of people who visit Hospital 1 - Number of people who both visit Hosp 1 and are not admitted) + (Number of people not admitted anywhere)
= (Number of people who visit Hosp 1 and are admitted) + (Number of people not admitted anywhere)
The number of people not admitted anywhere is the total number of people minus the total admitted people, or 22,252 - 4485 = 17,767. So we have:
Total = (1277) + (17,767)
= 19,044.
(e) A' n B' = the people who both do not visit hospital 1 AND are not admitted anywhere. This is equivalent to the people who are NOT either visiting hospital 1 or being admitted, or (A U B)' (by DeMorgan's Laws). So the total is
Total = Total number of people - [(Number of people visiting Hosp 1) + (Number of people admitted somewhere) - (Number of people both visiting Hosp 1 and admitted)]
(Note that this is just the total number of people minus the answer to part (c))
= 22,252 - [5292 + 4485 - 1277]
= 13,752.
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