I had an AP Statistics final exam today and there was one multiple choice question which stumped me. It was along the lines of the following:
- Two independent traffic lights
- The probability that both are red is $0.22$
- The probability that the first is red and the second is not is $0.33$
- What is the probability that the second traffic light is red?
Options:
A) $0.40$
B) $0.45$
C) $0.50$
D) $0.55$
E) $0.60$
I was told that the correct answer is $0.40$ but I couldn't figure out how to get there. Any tips? Thanks.
Best Answer
let $p$ and $q$ be the probability that the first and second lights are red respectively.
Since we have independence we are essentially told:
$pq=0.22$ and $p(1-q)=0.33$
dividing both equation we deduce: $\frac{q}{1-q}=\frac{2}{3}$.
this equation is solved readily: $\frac{q}{1-q}=\frac{2}{3}\iff 3q=2-2q\iff q=\frac{2}{5}$.