1. Eight hundred people were tested for immunity to the diseases tuberculosis, r
ID: 2983398 • Letter: 1
Question
1. Eight hundred people were tested for immunity to the diseases tuberculosis, rubella, and smallpox. Of the 800 people, 350 were found to have immunity to tuberculosis, 450 to rubella, 350 to smallpox, 150 to tuberculosis and rubella, 200 to rubell and smallpox, 250 to tuberculosis and smallpos, and 100 to tuberculosis, rubella, and not smallpox. How many people were found to have immunity to none of these diseases?
2. Of 100 cars tested at an inspection station, 9 had defective headlights; 8 defective brakes; 7 defective horns; 2 defective windshield wipers; 4 defective headlights and brakes; 3 defective headlights and horns; none defective brakes and windshield wipers; 1 defective horn and windshield wipers; 1 defective headlights, brakes, and horn; 1 defective headlights, horn, and windshield wipers; and none had any other combinations of defects. Find the number of cars that had at least one of the defefcts in the question.
Explanation / Answer
(a)
T - people who are immune to tuberculosis
R - people who are immune ro rubelia
S - people who are immune to smallpox
P- set of all people tested.
^ is used for intersection and ~ for complement set
n(P) = 800
n(T) = 350
n(R) = 450
n(S) = 350
n(T ^ R) = 150
n(R ^ S) = 200
n(S^T) = 250
n(T^R^~S) = 100
n(T^R^S) + n(T^R^~S) = n(T^R) = 150.
thus n(T^R^S) = 150 -100 = 50
no of people who are immune to none of the diseases
= n(P) - n(T U R U S)
= n(P) - (n(T) +n(R)+n(S)) + ( n(T^R)+n(R^S)+n(S^T))-n(R^S^T)
= 800 - 1150 + 600 - 50 = 200
(b)No of cars that had atleast one defect
= (9+8+7+2) - (4+3+1) + (1+1)
(inclusion exclusion principle n(A U B U C U D) = summation cardinality of sets taken one at a time - summation (cardinality of sets taken two at a time) + summation cardinality of sets taken three at a time - summation cardinality of sets taken four at a time)
= 20
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