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A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 4 % of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 10 % of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 18 % of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.

Part a) What is the probability that the driver is incorrectly classified as being over the limit?

Part b) What is the probability that the driver is correctly classified as being over the limit?

Part c) Find the probability that the driver gives a breathalyser test reading that is over the limit.

Part d) Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit.

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    This is a basic exercise on Bayes' Theorem. Please show us some of your own work so we can know why you need help. To start can you explain the second probability and find the third: $P(\text{Over limit}) = .18,\,P(\text{Fail test | Over limit}) = 1 - .10 = .90,\,P(\text{Over limit} \cap \text{Fail test})=\,??$ If you can, please edit that into your problem, do as much as you can beyond, and let us know if you need further help. – 2017-01-31

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The easiest way to answer all your questions is to think in terms of frequencies. Construct table with $100$ people, $82$ are below the limit, $18$ are above the limit. Out of the $82$, $82\cdot0.04=3.28$ will get reading above and $78.72$ will get reading below. Out of the $18$, $1.8$ will get reading below and $16.2$ will get reading above. That is: $$\begin{array}{ccc} &\text{test above}&\text{test below}\\ \text{consumption above}&16.2&1.8\\ \text{consumption below}&3.28&78.72\\ \end{array}$$

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    Nice tutorial approach. (+1) – 2017-01-31