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A college has 80 students in Year 12 - Edexcel - A-Level Maths Statistics - Question 3 - 2015 - Paper 1
Question 3
A college has 80 students in Year 12. 20 students study Biology 28 students study Chemistry 30 students study Physics 7 students study both Biology and Chemistry 11... show full transcript
Worked Solution & Example Answer:A college has 80 students in Year 12 - Edexcel - A-Level Maths Statistics - Question 3 - 2015 - Paper 1
Step 1
Draw a Venn diagram to represent this information.
Answer
To represent the given data in a Venn diagram, we would draw three intersecting circles for Biology, Chemistry, and Physics. We will start by labeling the numbers in the overlapping regions:
- 3 students represent those studying all three subjects.
- 7 students study both Biology and Chemistry, so 4 will be only in the Biology and Chemistry overlap section.
- 11 students study both Chemistry and Physics, meaning 8 are only in Chemistry and Physics.
- 5 students study both Physics and Biology; hence, 2 are only in this section.
- Finally, we put appropriate labels to ensure the regions sum up to the total number of students studying each subject.
Step 2
Find the probability that the student studies Chemistry but not Biology or Physics.
Answer
The number of students studying only Chemistry can be calculated as follows:
egin{align*}
ext{Total studying Chemistry} &= 28
ext{Students studying both Biology and Chemistry} &= 4
ext{Students studying both Chemistry and Physics} &= 8
ext{Students studying all three subjects} &= 3
ext{Students studying only Chemistry} &= 28 - (4 + 8 + 3) = 13
ext{Total students} &= 80
ext{Probability} &= \frac{13}{80} = 0.1625
ext{or } 16.25%
ext{Therefore, } P( ext{Chemistry and not Biology or Physics}) = 0.1625.
\end{align*}
Step 3
Find the probability that the student studies Chemistry or Physics or both.
Answer
To find this probability, we first find the total studying Chemistry and Physics:
egin{align*}
ext{Total studying Chemistry} &= 28
ext{Total studying Physics} &= 30
ext{Students studying both Chemistry and Physics} &= 8
P( ext{Chemistry or Physics}) & = P( ext{Chemistry}) + P( ext{Physics}) - P( ext{Chemistry and Physics})
&= \frac{28}{80} + \frac{30}{80} - \frac{8}{80}
&= \frac{50}{80} = 0.625
\end{align*}
Thus, the probability that the student studies Chemistry or Physics or both is 0.625.
Step 4
Given that the student studies Chemistry or Physics or both, find the probability that the student does not study Biology.
Answer
To calculate the probability that the student does not study Biology given they study Chemistry or Physics:
First, we determine the number of students who study Chemistry or Physics, which is already computed as:
- Total studying Chemistry or Physics = 50.
Now, from these, we need to exclude those who study Biology:
- Total studying Biology = 20
- Total in the overlap (includes those studying all three) = 3
- Non-Biology students from Chemistry or Physics = 50 - (4 + 8 + 3) = 35
Thus, the probability is:
egin{align*} P( ext{Not Biology | Chemistry or Physics}) &= \frac{35}{50} = 0.7 \end{align*}
Step 5
Determine whether studying Biology and studying Chemistry are statistically independent.
Answer
To determine independence, we check:
egin{align*}
P(B ext{ and } C) &= P(B) imes P(C)
\text{Where } P(B) &= \frac{20}{80} = 0.25
P(C) &= \frac{28}{80} = 0.35
\text{So } P(B ext{ and } C) &= \frac{7}{80}
\0.25 \times 0.35 &= 0.0875
\text{Now comparing: } P(B ext{ and } C) \neq 0.0875
\text{This implies, }\text{they are independent.}
\end{align*}