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14 A teacher in a college asks her mathematics students what other subjects they are studying - AQA - A-Level Maths Pure - Question 14 - 2018 - Paper 3
Question 14
14 A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students: 12 study physics 8 study g... show full transcript
Worked Solution & Example Answer:14 A teacher in a college asks her mathematics students what other subjects they are studying - AQA - A-Level Maths Pure - Question 14 - 2018 - Paper 3
Step 1
Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent.
Answer
To determine the independence of the events, we need to calculate the probabilities:
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Calculate (P(Physics)): [ P(Physics) = \frac{12}{24} = \frac{1}{2} ]
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Calculate (P(Geography)):
[ P(Geography) = \frac{8}{24} = \frac{1}{3} ] -
Calculate (P(Physics \cap Geography)):
[ P(Physics \cap Geography) = \frac{4}{24} = \frac{1}{6} ] -
Check if (P(Physics \cap Geography) = P(Physics) \times P(Geography)):
[ P(Physics) \times P(Geography) = \left(\frac{1}{2}\right) \times \left(\frac{1}{3}\right) = \frac{1}{6} ]
Since (P(Physics \cap Geography) = P(Physics) \times P(Geography)), we conclude that the events are independent.
Step 2
Calculate the probability that a student studies mathematics or biology or both.
Answer
Using the information provided, we can calculate the required probability:
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Let (P(M)) be the probability of studying mathematics and (P(B)) be the probability of studying biology: [ P(M) = \frac{1}{5}, \quad P(B) = \frac{1}{6} ]
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Calculate (P(B | M)) which is given as (\frac{3}{8}).
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Using the addition rule for probabilities: [ P(M \cup B) = P(M) + P(B) - P(M \cap B) ] Where:
[ P(M \cap B) = P(M) \times P(B | M) = \frac{1}{5} \times \frac{3}{8} = \frac{3}{40} ] -
Now plug the values into the addition rule:
[ P(M \cup B) = \frac{1}{5} + \frac{1}{6} - \frac{3}{40} ] -
Convert to a common denominator (40):
[ P(M \cup B) = \frac{8}{40} + \frac{6.67}{40} - \frac{3}{40} = \frac{11.67}{40} \approx \frac{7}{24} ]
Thus, the probability is approximately (\frac{7}{24}).