Question

A study was done to determine the efficacy of three different drugs—A, B, and C—in relieving headache pain. Over the period covered by the study, 50 subjects were given the chance to use all three drugs.

The following results were obtained:

21 reported relief from drug A

21 reported relief from drug B

29 reported relief from drug C

6 reported relief from both drugs A and B

13 reported relief from both drugs A and C

14 reported relief from both drugs B and C

42 reported relief from at least one of the drugs.

Note that some of the 21 subjects who reported relief from drug A may also have reported relief from drugs B or C. A similar occurrence may be true for the other data.

(a) How many people got relief from none of the drugs?

(b) How many people got relief from all three drugs?

(c) Let A be the set of all subjects who got relief from drug A, B the set of all subjects who got relief from drug B, and C the set of all subjects who got relief from drug C. Fill in the numbers for all eight regions of the following diagram. There are three intersecting circles that form a Venn diagram. The circles are labeled A, B, and C. Outside of the circles is labeled Sample of Students.

d) How many subjects got relief from A only?

Answer 1

Question (a)

Number of people got relief from none of the drugs = total number of people - people who releift from atleast one of drugs

= 50 - 42

= 8

Question (b)

n(ABC) = n(A) + n(B) + n(c) - n(AB) - n(BC) - n(CA) + n(ABC)

42 = 21 + 21 + 29 - 6 - 13 - 14 + n(ABC)

n(ABC) = 4

people got relief from all three drugs = 4

Question (c)

The venn diagram is as follows

The values in red represent that got relief from only A, only B, only C

The values in Green represent that got relief from A and B, B and C, A and C

The value in blue at the intersection of all three circles represent that got relief from A and B and C

First we will input the people who got relief from all the drugs as 4

Now the people who got relief only from drugs A and B = people who got relief from drugs A and B - people who got relief from all the drugs A, B,C

the people who got relief only from drugs A and B = 6 - 4

= 2

Likewise the people who got relief only from drugs B and C, from only drugs A and C are 10, 9

People who relief from only A = People who got relief from A - People who got relief from only A and B - People who got relief from only A and B and C - People who got relief from only A and C

People who got relief from only A = 21 - 2 - 4 - 9

= 6

Likewise the people who got relief from only B is 5 and people who got relief from only C is 6

Question (d)

subjects got relief from A only = 6 as calculated and shown in the above diagram

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A (2) 5