In a laboratory experiment, two independent sensors, $S_1$ and $S_2$, monitor a chemical reaction. Sensor $S_1$ detects a change with probability $p_1$ and $S_2$ with $p_2$. If the sensors are triggered only if at least one detects the change, but an alarm sounds only when both detect it, what is the probability that $S_1$ detected the change given that the alarm sounded?