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In a class of 30 students, 20 study Physics, 6 study Biology and 4 study both Physics and Biology - Leaving Cert Mathematics - Question b - 2013
Question b
In a class of 30 students, 20 study Physics, 6 study Biology and 4 study both Physics and Biology. A student is selected at random from this class. The events E and... show full transcript
Worked Solution & Example Answer:In a class of 30 students, 20 study Physics, 6 study Biology and 4 study both Physics and Biology - Leaving Cert Mathematics - Question b - 2013
Step 1
Represent the information on the Venn Diagram.
Answer
To represent the information on the Venn Diagram, we need to fill in the respective areas:
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Total students studying Physics = 20
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Students studying both Physics and Biology = 4
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Therefore, students studying only Physics = 20 - 4 = 16.
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Total students studying Biology = 6
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Students studying both Physics and Biology = 4
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Therefore, students studying only Biology = 6 - 4 = 2.
The Venn Diagram will appear as follows:
Physics Biology
[16] [2]
[4]
This indicates: 16 students study only Physics, 2 students study only Biology, and 4 students study both subjects.
Step 2
By calculating probabilities, investigate if the events E and F are independent.
Answer
To check if events E and F are independent, we calculate the probabilities:
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Calculate P(E ∩ F) (students studying both subjects):
- From the Venn Diagram, P(E ∩ F) = Number of students studying both Physics and Biology / Total students = ( \frac{4}{30} )
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Calculate P(E) (students studying Physics):
- P(E) = Total Physics students / Total students = ( \frac{20}{30} )
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Calculate P(F) (students studying Biology):
- P(F) = Total Biology students / Total students = ( \frac{6}{30} )
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Calculate P(E) × P(F):
( P(E) \times P(F) = \frac{20}{30} \times \frac{6}{30} = \frac{120}{900} = \frac{4}{30} )
Since ( P(E ∩ F) = P(E) \times P(F) ), we conclude that E and F are independent events.