A drug trial had 649 participants. A survey was taken to determine how many had dizziness as a side effect, how many had anxiety, and how many had fatigue. The Venn diagram below s... A drug trial had 649 participants. A survey was taken to determine how many had dizziness as a side effect, how many had anxiety, and how many had fatigue. The Venn diagram below shows the results. (Each number gives the number of participants who fall into that Venn diagram category.) (a) How many of the participants had fatigue but didn't have dizziness? (b) How many of the participants had neither anxiety nor fatigue? (c) How many of the participants had exactly one of the three side effects?
Understand the Problem
The question presents a Venn diagram showing the number of participants in a drug trial who experienced different side effects: dizziness, anxiety, and fatigue. The question asks us to extract information from the Venn diagram to answer three questions. (a) the number of participants who had fatigue but didn't have dizziness. (b) the number of participants who had neither anxiety nor fatigue. (c) the number of participants who had exactly one of the three side effects.
Answer
(a) 156 (b) 182 (c) 187
Answer for screen readers
(a) 156 participants had fatigue but didn't have dizziness. (b) 182 participants had neither anxiety nor fatigue. (c) 187 participants had exactly one of the three side effects.
Steps to Solve
- Find participants with fatigue but without dizziness
We are looking for participants who experienced fatigue but did not experience dizziness. In the Venn diagram, this corresponds to the regions of the 'fatigue' circle that do not overlap with the 'dizziness' circle. This includes the section of fatigue which overlaps with anxiety only $54$ and the section of fatigue that does not overlap with any other category $102$. $$ 54 + 102 = 156 $$
- Find participants with neither anxiety nor fatigue
We need to find the number of participants who did not experience anxiety and did not experience fatigue. In the Venn diagram, this corresponds to the region outside of both the 'anxiety' and 'fatigue' circles. This includes two regions: participants only experienced dizziness $47$ and those who experienced dizziness, but dizziness, anxiety, and fatigue $135$. $$ 47 + 135 = 182 $$
- Find participants with exactly one side effect
We are looking for the number of participants who experienced only one of the three side effects. In the Venn diagram, this corresponds to the regions of each circle that do not overlap with any other circle. This includes participants who only experienced dizziness $47$, participants who only experienced anxiety $38$, and participants who only experienced fatigue $102$. $$ 47 + 38 + 102 = 187 $$
(a) 156 participants had fatigue but didn't have dizziness. (b) 182 participants had neither anxiety nor fatigue. (c) 187 participants had exactly one of the three side effects.
More Information
Venn diagrams are very useful for visualizing set relationships and quickly extracting information. In this case, each circle represents a set of people that experienced one of the three side effects. The overlapping regions represent the intersections of these sets. The area outside of the circles represents all the people that were not in any of the original three sets.
Tips
A common mistake when working with Venn diagrams is misinterpreting the overlapping regions. For example, including the intersection between fatigue and dizziness when looking for people who experienced only fatigue would be incorrect.
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