In a medical study involving 300 clinical trial participants, researchers recorded the occ… — GMAT

In a medical study involving $300$ clinical trial participants, researchers recorded the occurrence of three common side effects: fatigue, nausea, and insomnia. A total of $160$ participants reported fatigue, $140$ reported nausea, and $120$ reported insomnia. Additionally, $60$ participants reported both fatigue and nausea, $50$ reported both fatigue and insomnia, and $55$ reported both nausea and insomnia. If $30$ participants reported none of these three side effects, how many participants reported exactly two of these side effects?

Cevap: 120 participants

Cevap

120 participants reported exactly two of the side effects.

To calculate the number of participants with exactly two side effects, first find the total number of participants experiencing at least one side effect (

300 - 30 = 270

). Next, using the inclusion-exclusion formula

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, we find that

x = 15

participants experienced all three side effects. Finally, subtracting

15

from each pairwise overlap gives the counts for exactly two side effects:

45

(Fatigue & Nausea only),

35

(Fatigue & Insomnia only), and

40

(Nausea & Insomnia only). Summing these regions yields

45 + 35 + 40 = 120

Adım Adım Çözüm

Calculate the total number of participants who experienced at least one side effect

270 participants

Subtracting the 30 participants who reported no side effects from the total study size of 300 gives

|F \cup N \cup I| = 300 - 30 = 270

Determine the number of participants who experienced all three side effects

15 participants

Applying inclusion-exclusion gives

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, simplifying to

270 = 255 + x

, so

x = 15

Calculate the sum of participants experiencing exactly two side effects

120 participants

The sum of pairwise intersections (

60 + 50 + 55 = 165

) counts individuals with all three side effects three times. Subtracting

3 \times 15 = 45

from

165

yields

120

Anahtar Kavram

Three-Set Overlapping Sets and Inclusion-Exclusion Principle

Tahmini Süre:2m 0s

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