Question

Police plan to enforce speed limits by using radar system at $3$ different locations within the city limits. The radar system at each of these locations are operated for $40$%, $30$% and $20$% of the time. If a person who is speeding on his way to work has probabilities of $0.2,0.1$ and $0.5$ respectively of passing throught these locations, what is the probability that he will be fined?

Answer 1

Let these three locations be $L_1,L_2$ and $L_3$.

Now, consider each individual location (say $L_1$). The probability that he will pass through this location is $0.2$. Also, to get fined while speeding, the radar has to be in operating condition. The probability that the the radar is operated at this location is $0.4$. SInce, both these events are mutually inclusive,

$P\left(L_1\right)=0.2\times 0.4=0.08$

Similarly,

$P\left(L_2\right)=0.1\times 0.3=0.03$

$P\left(L_3\right)=0.5\times 0.2=0.10$

Since, the probabilities at these three locations are mutually exclusive, joint probability that he is fined is,

$P=0.08+0.03+0.10=0.21$

Hence, this is the is the required probability.