Question

Let \(E\) and \(F\) be two independent events. The probability that both \(E\) and \(F\) happen is \(\dfrac1{12}\) and the probability that neither \(E\) nor \(F\) happens is \(\dfrac12\), then a value of \(\dfrac{P(E)}{P(F)}\) is :

• A. \(\dfrac43\)
• B. \(\dfrac32\)
• C. \(\dfrac13\)
• D. \(\dfrac5{12}\)

Hint:

Independence: \(pq=\frac1{12}\) and \(P(E'\cap F')=(1-p)(1-q)=\frac12\).
\(1-(p+q)+pq=\frac12\Rightarrow p+q=1+\frac1{12}-\frac12=\frac7{12}\).
\(p,q\) are roots of \(t^2-\frac7{12}t+\frac1{12}=0\Rightarrow12t^2-7t+1=0\Rightarrow t=\frac13,\frac14\).
So \(\frac{P(E)}{P(F)}=\frac{1/3}{1/4}=\frac43\) or \(\frac{1/4}{1/3}=\frac34\). Of the given options, \(\frac43\) appears.
Answer: (1) \(\frac43\)

Answer 1

The Correct Option is A

Let \(P(E)=p,\ P(F)=q\); use \(pq=\frac1{12}\) and \((1-p)(1-q)=\frac12\) to get \(p+q\).