Answer
One-twelfth of graduates got into medical school with an MCAT lower than twenty-seven and a GPA below 3.5.
Work Step by Step
Let A be the event that graduates got into medical school with an MCAT score of twenty-seven or higher.
Let B be the event that graduates got into medical school with a GPA of 3.5 or higher.
We have given that:
\[N(S)=1200\]
\[N(A)=1000\]
\[N(B)=400\]
\[N(A\cap B)=300\]
We have to find the number of graduates who got into medical school with an MCAT lower than twenty-seven and a GPA below 3.5; that means \[N({{A}^{C}}\cap {{B}^{C}})\].
By De Morgan’s law, we can write \[({{A}^{C}}\cap {{B}^{C}})\]as:
\[({{A}^{C}}\cap {{B}^{C}})={{(A\cup B)}^{C}}\]
We find first \[N(A\cup B)\]:
\[\begin{align}
& N(A\cup B)=N(A)+N(B)-N(A\cap B) \\
& N(A\cup B)=1000+400-300 \\
& N(A\cup B)=11000 \\
\end{align}\]
Then,
\[\begin{align}
& N{{(A\cup B)}^{C}}=N(S)-N(A\cup B) \\
& N{{(A\cup B)}^{C}}=1200-1100 \\
& N{{(A\cup B)}^{C}}=100 \\
\end{align}\]
Therefore, the proportion of those twelve hundred graduates got into medical school with an MCAT lower than twenty-seven and a GPA below 3.5 is calculated as:
\[\begin{align}
& =\frac{N{{(A\cup B)}^{C}}}{N(S)} \\
& =\frac{100}{1200} \\
& =\frac{1}{12} \\
\end{align}\]
The required proportion is one- twelfth.
