The chances of three independent events X, Y, Z occurring are \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{1}{4}\) respectively....

JAMB 1993

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. \(\frac{1}{4}\)

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Correct Answer: Option C

Explanation

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 \(\frac{1}{2}\)

= \(\frac{1}{12}\)