Question 905·Hard·Math

The first “given that” keeps students not in Art. Subtract the $90$ Art participants from all $210$ surveyed students. Type $210-90$ in Desmos; it prints $120$. That is the group used by the first probability, not all $210$ students.

Of those $120$ students, the conditional probability $\frac{1}{4}$ says one-fourth participate in Music. Because none participates in Art, these are Music-only students. Type $120/4$ in Desmos; it prints $30$.

Take all $90$ Art students and add the $30$ Music-only students. Type $90+30$ in Desmos; it prints $120$ students in at least one program. Don't add the students in both again: they're already among the $90$ Art students.

Let $b$ count students in the overlap, meaning both programs. The $90$ Art students include that overlap, so subtract $b$ to get Art only:

Exactly one program means Art only or Music only, never both. Add those two counts to get the number the second probability describes:

The second “given that” keeps only students in at least one program, a group of $120$. A conditional probability puts the count described in step 5 over that group's total, so:

Type $\frac{(90-x_1)+30}{120}\sim\frac{3}{5}$ in Desmos. Here $x_1$ replaces $b$, and $\sim$ tells Desmos to fit the two sides. Under PARAMETERS, it shows $x_1=48$. Since $b$ counts students in both Art and Music, $48$ surveyed students participate in both. Grid in 48.