The chances of three independent events X, Y, Z occurring are \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{1}{4}\) respectively. What are the chances of Y and Z only occurring?
A
\(\frac{1}{8}\)
B
\(\frac{1}{24}\)
C
\(\frac{1}{12}\)
D
\(\frac{1}{4}\)
Verified Correct Answer: C.
Step-by-Step Mathematical Solution
Chance of x = \(\frac{1}{2}\)
Change of Y = \(\frac{2}{3}\)
Chance of Z = \(\frac{1}{4}\)
Chance of Y and Z only occurring
= Pr (Y ∩ Z ∩ Xc)
where Xc = 1 - Pr(X)
1 - \(\frac{1}{2}\) = \(\frac{1}{2}\)
= Pr(Y) x Pr(Z) x Pr(Xc)
= \(\frac{2}{3}\) x \(\frac{1}{4}\) x \(\frac{1}{2}\)
= \(\frac{1}{12}\)
Source: Myschool Past Questions Archive (Record #31290)