A building has $!48$! one- and two-bedroom apartments to rent, some wi

Here is the full "GMAT Jujitsu" for this question:

First, there is a classic trap embedded in this question. The problem tricks you into thinking that you have two "separate" probabilities: (1) a 40 out of 48 chance that an apartment will have two bedrooms and (2) a 30 out of 48 chance an apartment will have a balcony. The trap is to get you to think that we simply need to multiply these probabilities by each other to calculate the combined probability. (\(\frac{40}{48}*\frac{30}{48}=\frac{1200}{2304}>50\%\)) This would result in answer choice C, which is NOT the right answer. The problem with this approach is that the probabilities are NOT independent of each other, but are instead overlapping. You can't just multiply probabilities together in this case.

In reality, we have two mutually exclusive groups (2-bedrooms vs. 1-bedroom and balconies vs. no balconies.) If you can organize your thoughts around these mutually-exclusive groups by creating a specialized table, the solution to this problem naturally unfolds. (I call a such tables "Matrix Boxes" in my classes.)

Here is what the blank table might look like:
\[
\begin{matrix}
& \text{Balconies} & \text{No Balconies} & \text{Totals} \\
\text{1-Bedroom} & \text{____} & \text{____} & \text{____} \\
\text{2-Bedrooms} & \text{____} & \text{____} & \text{____} \\
\text{Totals} & \text{____} & \text{____} & \text{____}
\end{matrix}
\]
The question tells us that there are 48 total apartments. Let's start there:
\[
\begin{matrix}
& \text{Balconies} & \text{No Balconies} & \text{Totals} \\
\text{1-Bedroom} & \text{____} & \text{____} & \text{____} \\
\text{2-Bedrooms} & \text{____} & \text{____} & \text{____} \\
\text{Totals} & \text{____} & \text{____} & \text{48}
\end{matrix}
\]
Statement #1 tells us that 40 of the 48 apartments have 2-bedrooms and 30 of the 48 have balconies. Since both of these numbers are more than 24 (half of the total 48), these statements are insufficient by themselves. After all, we want to have sufficient information to determine if the probability of getting both 2-bedrooms and a balcony is greater than 50%. If either of the two conditions were less than 50% of the apartments, then we would have our answer immediately. (In this hypothetical case, there would be no way to have more than one answer to the question, and the statements would be sufficient.) Since it is possible to have greater than 50%, but it is also possible to have less than 50%, we can eliminate answers A, B, and D.

Once we combine the statements, here is what our Matrix Box would look like:
\[
\begin{matrix}
& \text{Balconies} & \text{No Balconies} & \text{Totals} \\
\text{1-Bedroom} & \text{____} & \text{____} & \text{8} \\
\text{2-Bedrooms} & \text{____} & \text{____} & \text{40} \\
\text{Totals} & \text{30} & \text{18} & \text{48}
\end{matrix}
\]
And yet, we still do not know the numbers on the inside of the table. With this information, it is possible that all of the 2-bedroom apartments have balconies, resulting in this combination:
\[
\begin{matrix}
& \text{Balconies} & \text{No Balconies} & \text{Totals} \\
\text{1-Bedroom} & \text{0} & \text{8} & \text{8} \\
\text{2-Bedrooms} & \text{30} & \text{22} & \text{40} \\
\text{Totals} & \text{30} & \text{18} & \text{48}
\end{matrix}
\]
The chance of having a 2-bedroom apartment with a balcony would therefore be \(\frac{30}{48}=\frac{5}{8}\) > \(50\%\).
But it is also possible that all of the 1-bedroom apartments have balconies, resulting in this combination:
\[
\begin{matrix}
& \text{Balconies} & \text{No Balconies} & \text{Totals} \\
\text{1-Bedroom} & \text{8} & \text{0} & \text{8} \\
\text{2-Bedrooms} & \text{22} & \text{18} & \text{40} \\
\text{Totals} & \text{30} & \text{18} & \text{48}
\end{matrix}
\]
In this case, the chance of having a 2-bedroom apartment with a balcony would therefore be \(\frac{22}{48}=\frac{11}{24}\) < \(50\%\).

Since we can have two answers to the question "is the probability greater than 1/2 that the apartment has two bedrooms and a balcony?", the answer is E.

Thanks AaronPond for you sufficient and detailed explanation.

(1) Of the apartments, 40 have two bedrooms. insufic

……one……two……total
bal……………?…………
not………………………
……(8)……(40)………48

(2) Of the apartments, 30 have a balcony. insufic

……one……two……total
bal……………?………(30)
not…………………….(18)
………………………….48

(1 & 2) insufic

MIN…one……two……total
bal…………[22]………(30)
not…………[18]…….(18)
……(8)……(40)………48

MIN probability 'two with bal' = 22/48 = 11/24 < 1/2

MAX…one……two……total
bal…………[30]………(30)
not…………[10]…….(18)
……(8)……(40)………48

MAX probability 'two with bal' = 30/48 = 15/24 > 1/2

Ans (E)