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14 A teacher in a college asks her mathematics students what other subjects they are studying - AQA - A-Level Maths Mechanics - 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 geogra... 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 Mechanics - 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 if the events are independent, we need to calculate the probabilities:
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Calculate (P(\text{Physics})) and (P(\text{Geography})):
- (P(\text{Physics}) = \frac{12}{24} = \frac{1}{2})
- (P(\text{Geography}) = \frac{8}{24} = \frac{1}{3})
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Calculate the probability of both events occurring:
- (P(\text{Physics} \cap \text{Geography}) = \frac{4}{24} = \frac{1}{6})
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Check if (P(\text{Physics}) \times P(\text{Geography}) = P(\text{Physics} \cap \text{Geography})):
- (P(\text{Physics}) \times P(\text{Geography}) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6})
- Since (P(\text{Physics} \cap \text{Geography}) = \frac{1}{6}), the events are independent.
Step 2
Calculate the probability that a student studies mathematics or biology or both.
Answer
Let (P(M)) be the probability of studying mathematics, (P(B)) be the probability of studying biology, and (P(M \cap B)) be the probability of studying both:
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Calculate (P(M) = \frac{1}{5}) and (P(B) = \frac{1}{6}).
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To find (P(M \cap B)), we use the conditional probability formula:
- (P(B | M) = \frac{P(M \cap B)}{P(M)}) therefore (P(M \cap B) = P(B | M) \times P(M))
- Substitute (P(B | M) = \frac{3}{8}) and (P(M) = \frac{1}{5}):
- (P(M \cap B) = \frac{3}{8} \times \frac{1}{5} = \frac{3}{40})
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Use the addition rule to calculate (P(M \cup B)):
- (P(M \cup B) = P(M) + P(B) - P(M \cap B))
- Substitute the values:
- (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} = \frac{35}{120} = 0.2917)
- Thus, the probability that a student studies mathematics or biology or both is approximately 0.2917.