At a technology festival, 180 students could attend any combination of three workshops: programming (), electronics (), and design (). The attendance data are as follows:
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92 students attended .
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80 students attended .
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70 students attended .
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38 students attended both and , including any students who also attended .
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30 students attended both and , including any students who also attended .
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26 students attended both and , including any students who also attended .
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14 students attended all three workshops.
A student is selected at random from the students who attended at least two of the workshops. What is the probability that the selected student attended both and ?
For conditional probability with overlapping groups, first identify exactly which students belong in the condition. When pairwise counts include a three-way overlap, subtract the three-way overlap from each pair before adding categories together. Then use the count meeting both the condition and the requested event as the numerator, divided by the count meeting the condition as the denominator.
Hints
Interpret the pairwise counts
The count for each pair of workshops includes students who may have attended the third workshop too.
Avoid double-counting
Subtract the all-three-workshops count from each pairwise count to find how many students attended exactly that pair.
Use the condition as the denominator
For “at least two workshops,” include the three exactly-two-workshop groups and the all-three-workshops group in the denominator.
Desmos Guide
Use counting first
Direct counting is faster than graphing for this question. In Desmos, enter ((38-14)+(30-14)+(26-14)+14) to verify the number of students who attended at least two workshops.
Check the probability
Enter 38/66 and reduce the displayed fraction or decimal. The numerator is 38 because the and count already includes students who attended all three workshops.
Step-by-step Explanation
Find the numbers who attended exactly two workshops
Each pairwise count includes the 14 students who attended all three workshops. Therefore:
Count the students in the conditional group
Students who attended at least two workshops consist of the three exactly-two-workshop groups and the all-three-workshops group:
Form the conditional probability
All 38 students who attended both and belong in the conditional group because they attended at least two workshops. Thus,