Question

The students in Muggle Studies class (Muggles are non-magic people) organize a party for Hogwarts students midway through a semester. Of the students attending the party, 40% are Gryffindor students, 30% are Hufflepuff students, 20% are Ravenclaw students, and 10% are Slytherin students. Magic is not allowed at this party, but some mischievous students still use it. Only 10% of Gryffindor students at the party use magic. Same can be said about 20% of Hufflepuffs at the party. Among Ravenclaw students at the party, 40% use magic. Predictably, 70% of Slytherins at the party use magic.
a) What proportion of the students ate the party use magic?
b) If a student at the party does not use magic, what is the probability that this student is from Gryffindor?
c) If a student at the party does use magic, what is the probability that this student is not from Gryffindor?
d) What proportion of the students who are not from Gryffindor do not use magic?
e) What proportion of the students at the party either are from Gryffindor or do not use magic, or both?

Answer 1

Answer: (a) 1/4    (b) 8/15     (c) 5/12     (d) 13/20      (e) 79/100

Step-by-step explanation:

Create a table of values (below) as it will be easier to do the calculations.

$\begin{array}{l|c|c|c}&\underline{\text{Total at party}}&\underline{\text{Used magic}}&\underline{\text{Did NOT use magic}}\\\text{Gryffindor}&0.40&0.4(0.1)=0.04&0.4(0.9)=0.36\\\text{Hufflepuff}&0.30&0.3(0.2)=0.06&0.3(0.8)=0.24\\\text{Ravenclaw}&0.20&0.2(0.4)=0.08&0.2(0.6)=0.12\\\underline{\text{Slytherin}}&\underline{\qquad 0.10\qquad}&\underline{0.1(0.7)=0.07}&\underline{0.1(0.3)=0.03}\\\text{Totals}&1.00&0.25&0.75\end{array}$

$a)\quad \dfrac{\text{Used magic}}{\text{Total at party}}=\dfrac{0.25}{1.00}=\large\boxed{\dfrac{1}{4}}$

$b)\quad \dfrac{\text{Total Gryffindor}}{\text{Does NOT use magic}}=\dfrac{0.40}{0.75}=\large\boxed{\dfrac{8}{15}}$

$c)\quad \dfrac{\text{Uses magic}}{\text{Total NOT Gryffindor}}=\dfrac{0.25}{1.00-0.40}=\dfrac{0.25}{.60}=\large\boxed{\dfrac{5}{12}}$

$d)\quad \dfrac{\text{Does NOT use magic-NOT Gryffindor}}{\text{Total NOT Gryffindor}}=\dfrac{0.75-0.36}{1.00-0.40}=\dfrac{0.39}{0.60}=\large\boxed{\dfrac{13}{20}}$

$e)\quad \dfrac{\text{Gryffindor + Do NOT use magic}}{\text{Total at party}}=\dfrac{0.40+0.75-0.36}{1.00}=\large\boxed{\dfrac{79}{100}}$