The police department in a city plans to enforce the speed limit by using radar

Question

The police department in a city plans to enforce the speed limit by using radar traps at four locations. During morning rush hour exactly one trap at one location is operated. The radar trap will be operated at location L1, L2, L3, and L4 with probabilities 0.4, 0.3, 0.2, and 0.3, respectively. Samantha passes through all four locations each morning and speeds the entire way, but even if the trap is operating, she will not always get at ticket. If the trap is operating at location L1, the probability that Samantha will be given at ticket is 0.6. The similar probabilities that Samantha will be given at ticket if the trap is operating at location L2, L3, or L4 are 0.5, 0.9, and 0.6, respectively.

a.       Compute the probability Samantha will receive a speeding ticket one morning.

b.      If Samantha received a speeding ticket while speeding to school, what is the probability that the radar trap was operating at location L2?

Explanation / Answer

The police department in a city plans to enforce the speed limit by using radar traps at four locations. During morning rush hour exactly one trap at one location is operated. The radar trap will be operated at location L1, L2, L3, and L4 with probabilities 0.4, 0.3, 0.2, and 0.3, respectively. Samantha passes through all four locations each morning and speeds the entire way, but even if the trap is operating, she will not always get at ticket. If the trap is operating at location L1, the probability that Samantha will be given at ticket is 0.6. The similar probabilities that Samantha will be given at ticket if the trap is operating at location L2, L3, or L4 are 0.5, 0.9, and 0.6, respectively.

Bayes’ theorem used.

a.       Compute the probability Samantha will receive a speeding ticket one morning.

P( L1)=0.4    P(L2)=0.3    P(L3)=0.2     P(L4)=0.3

P( T/L1)=0.6    P(T/L2)=0.5    P(T/L3)=0.9     P(T/L4)=0.6

P(T) = P( T/L1)* P( L1)+ P(T/L2)* P(L2)+ P(T/L3)* P(L3)*+ P(T/L4)* P(L4)

=0.6*0.4 + 0.5*0.3 + 0.9*0.2 + 0.6*0.3

=0.75

b.      If Samantha received a speeding ticket while speeding to school, what is the probability that the radar trap was operating at location L2?

P(L2/T)= P(T/L2)* P(L2) / P(T)

= 0.5*0.3 / 0.75

= 0.2

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