Question

Please show work as I have a fundamental misunderstanding of this material

The State Police are trying to crack down on speeding on a particular portion of the Massachusetts Turnpike. To aid in this pursuit, they have purchased a new radar gun that promises greater consistency and reliability. Specifically, the gun advertises ± one-mile-per-hour accuracy 85% of the time; that is, there is a 0.85 probability that the gun will detect a speeder, if the driver is actually speeding. Assume there is a 2% chance that the gun erroneously detects a speeder even when the driver is below the speed limit. Suppose that 82% of the drivers drive below the speed limit on this stretch of the Massachusetts Turnpike.

a. What is the probability that the gun detects speeding and the driver was speeding?

b. What is the probability that the gun detects speeding and the driver was not speeding?

c. Suppose the police stop a driver because the gun detects speeding. What is the probability that the driver was actually driving below the speed limit?

Answer 1

Let X denote the event that the driver is actually speeding. P(X) = 1-0.82 = 0.18
Let Y denote the event that the speed gun detects a speedster, Then
Probability that the gun will detect a speeder if the driver is actually speeding: P(Y | X) = 0.85
Probability that the gun erroneously detects a speeder: P(Y | X^c) = 0.02 , where X^c means that the driver is below the speed limit and P(X^c) = 1- P(X) = 1 - 0.18 = 0.82.

a.
Probability that the gun detects speeding and the driver was speeding: P(Y ∩ X)
P(Y ∩ X) = P(X) . P(Y | X) = 0.18 x 0.85 = 0.153

b.
Probability that the gun detects speeding and the driver was not speeding: P(Y ∩ X^c)
P(Y ∩ X^c) = P(X^c) . P(Y | X^c) = 0.82 x 0.02 = 0.0164

c.
Given police stops a driver because the gun detects speeding, probability that the driver was actually driving below the speed limit: P(X^c | Y)
Using conditional probability,
P(X^c | Y) = P(X^c ∩ Y) / P(Y)
P(X^c ∩ Y) = 0.0164 from part b.
Calculate P(Y) using totl law of probability
P(Y) = P(X).P(Y | X) + P(X^c).P(Y | X^C)
P(Y) = (0.18 x 0.85) + (0.82 x 0.02) = 0.1694

P(X^c | Y) = P(X^c ∩ Y) / P(Y)
P(X^c | Y) = 0.0164 / 0.1694 = 0.0968