A drug trial had 603 participants. A survey was taken to determine how many had nausea as a side effect, how many had dizziness, and how many had fatigue. The Venn diagram below sh... A drug trial had 603 participants. A survey was taken to determine how many had nausea as a side effect, how many had dizziness, and how many had fatigue. The Venn diagram below shows the results. (a) How many of the participants had exactly two of the three side effects? (b) How many of the participants didn't have dizziness? (c) How many of the participants had neither fatigue nor nausea, but did have dizziness?
Understand the Problem
The question presents a Venn diagram showing the number of participants in a drug trial who experienced various side effects (nausea, dizziness, and fatigue). It asks us to determine the number of participants who had exactly two of the three side effects, the number of participants that didn't have dizziness, and the number of participants that had neither fatigue nor nausea, but did have dizziness.
Answer
(a) 142 (b) 241 (c) 119
Answer for screen readers
(a) 142 participants (b) 241 participants (c) 119 participants
Steps to Solve
- Find the number of participants with exactly two side effects
Identify the regions in the Venn diagram where only two circles overlap. These numbers represent participants with exactly two side effects. In the Venn diagram, these numbers are 63, 31, and 48. Then, we calculate the sum of these values. $$ 63 + 31 + 48 = 142 $$
- Find the number of participants without dizziness
To find the number of participants who didn't have dizziness, sum the numbers in the regions that are not within the 'dizziness' circle. This includes the regions representing 'nausea only' (32), 'fatigue only' (152), 'nausea and fatigue only' (57), and 'no side effects' (which we assume is 0 since it is not shown). Therefore, the sum is written as follows:
$$ 32 + 152 + 57 = 241 $$
- Find the number of participants with dizziness but neither fatigue nor nausea
Identify the region in the Venn diagram representing participants who had dizziness but neither fatigue nor nausea. This is the part of the 'dizziness' circle that doesn't overlap with either the 'fatigue' or 'nausea' circles. In the Venn diagram, this number is 119.
(a) 142 participants (b) 241 participants (c) 119 participants
More Information
Venn diagrams are a useful tool for visualizing set relationships and solving counting problems. Ensuring you identify the correct regions of the diagram is key to solving the problem correctly.
Tips
A common mistake is to include the number of participants that had all three side effects when calculating the number of participants that had exactly two side effects. Also, the numbers in the regions where only one circle applies (32, 119, 152) may mix up students. Another mistake is misreading the Venn diagram and including overlapping sections when they shouldn't be.
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