Question 1027797: Among the 78 doctors on the staff of a hospital, 64 carry malpractice insurance, 36 are surgeons, and 34 of the surgeons carry malpractice insurance. If one of these doctors is randomly chosen by lot to represent the hospital staff at an A.M.A. convention, what is the probability that the one chosen is not a surgeon and does not carry malpractice insurance? .
b)Given p(A)=0.59, p(B)= 0.30, and p(A∩B)=0.21. find
i) p(A∪B)
ii) p( A∩ (B ) ̅)
iii) p((A ) ̅ ∪ (B ) ̅)
iv) p( (A ) ̅∩ B ̅)
Answer by mathmate(429)   (Show Source):
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Question:
Among the 78 doctors on the staff of a hospital, 64 carry malpractice insurance, 36 are surgeons, and 34 of the surgeons carry malpractice insurance. If one of these doctors is randomly chosen by lot to represent the hospital staff at an A.M.A. convention, what is the probability that the one chosen is not a surgeon and does not carry malpractice insurance? .
b)Given p(A)=0.59, p(B)= 0.30, and p(A∩B)=0.21. find
i) P(A∪B)
ii) P( A∩ B̅)
iii) P(A̅ ∪ B̅)
iv) P( A̅∩ B̅)
Solution:
(to simplify typography, will use ~A to mean complement of A)
The solution of this problem is mainly based on the following relations:
1. P(~A∩~B)=P(~(A∪B)) [de Morgan's law]
2. P(A∪B)=P(A)+P(B)-P(A∩B) [relation union/intersection]
3. P(~A)=1-P(A) [def of complement]
4. P(A∩~B)=P(A-B)=P(A)-P(A∩B) [ def. of difference ]
Using |A|=cardinality (size) of set A
(a)
Define events:
I=carry insurance
S=surgent
Given:
P(I)=64/78
P(S)=36/78
P(I∩S)=34/78
Using rule (2) above
P(I∪S)=(64+36-34)/78=66/78
With the basic information, we can now calculate
using rule (1) and (3) above
P(~S∩~I)
=P(~(S∪I)) [ rule (1), de Morgan's law ]
=1-P(S∪I) [ rule (3), definition of complement ]
=1-66/78
=12/78
(b) Given P(A)=0.59, P(B)=0.30, P(A∩B)=0.21,
(i) P(A∪B)
=P(A)+P(B)-P(A∩B) [ using (2) ]
= 0.68
(ii) P(A∪~B)
=P(A)-P(A∩B) [ (4) def. of difference ]
=0.59-0.21
=0.38
(iii) P(~A∪~B)
=P(~(A∩B)) [ de Morgan ]
=1-P(A∩B) [ (3) def. of difference ]
=0.79
(iv) P(~A∩~B)
=P(~(A∪B)) [ de Morgan ]
=1-P(A∪B)) [def. complement]
=1-0.68
=0.32
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