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: 7 % 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.

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

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

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

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

To answer these questions, we can use conditional probability and Bayes' theorem. Let's define the following events:

A = Driver is incorrectly classified as being over the limit
B = Driver is correctly classified as being over the limit
C = Driver gives a breathalyser test reading that is over the limit
D = Driver is under the legal limit, given the breathalyser reading is also below the limit

We are given the following probabilities:
P(A|~C) = 7% = 0.07 (Probability of being incorrectly classified as over the limit given the breathalyser reading is below the limit)
P(B|C) = 90% = 0.90 (Probability of being correctly classified as over the limit given the breathalyser reading is over the limit)
P(C) = 18% = 0.18 (Probability of giving a breathalyser test reading that is over the limit)
P(~C) = 100% - P(C) = 1 - 0.18 = 0.82 (Probability of not giving a breathalyser test reading that is over the limit)

Now, let's find the answers to the given questions:

1. What is the probability that the driver is incorrectly classified as being over the limit?
Using Bayes' theorem: P(A) = P(A|~C) * P(~C) / P(~C) = 0.07 * 0.82 / 0.82 = 0.07

So, the probability that the driver is incorrectly classified as being over the limit is 0.07.

2. What is the probability that the driver is correctly classified as being over the limit?
Using Bayes' theorem: P(B) = P(B|C) * P(C) / P(C) = 0.90 * 0.18 / 0.18 = 0.90

So, the probability that the driver is correctly classified as being over the limit is 0.90.

3. Find the probability that the driver gives a breathalyser test reading that is over the limit.
This is already given as P(C) = 0.18

So, the probability that the driver gives a breathalyser test reading that is over the limit is 0.18.

4. Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit.
Using Bayes' theorem: P(D) = P(~C) * P(D|~C) / P(~C) = (1 - P(C)) * (1 - P(B|~C)) / (1 - P(C)) = 0.82 * (1 - 0.10) / 0.82 = 0.90

So, the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit is 0.90.

To solve this problem, we can use conditional probabilities. Let's define the following events:

A: The driver is actually over the legal alcohol limit.
B: The breathalyser test reading is over the limit.

We are given the following probabilities:

P(A) = 0.18 (18% of drivers are over the limit)
P(B|¬A) = 0.07 (7% of drivers who are not over the limit give a reading above the limit)
P(¬B|A) = 0.10 (10% of drivers who are over the limit give a reading below the limit)

Now let's calculate the probabilities step by step:

1. Probability of incorrect classification (driver incorrectly classified as over the limit):
P(¬A|B) = P(¬A) * P(B|¬A) / P(B)
P(¬A) = 1 - P(A) = 1 - 0.18 = 0.82
P(B) = P(B|¬A) * P(¬A) + P(B|A) * P(A) = 0.07 * 0.82 + (1 - 0.10) * 0.18 = 0.1426

P(¬A|B) = 0.82 * 0.07 / 0.1426 ≈ 0.4025

So, the probability that the driver is incorrectly classified as being over the limit is approximately 0.4025.

2. Probability of correct classification (driver correctly classified as over the limit):
P(A|B) = P(A) * P(B|A) / P(B)
P(A) = 0.18 (given)
P(B) = 0.1426 (calculated in step 1)

P(A|B) = 0.18 * (1 - 0.10) / 0.1426 ≈ 0.1266

So, the probability that the driver is correctly classified as being over the limit is approximately 0.1266.

3. Probability that the driver gives a breathalyser test reading that is over the limit:
P(B) = P(B|¬A) * P(¬A) + P(B|A) * P(A)
P(B) = 0.07 * 0.82 + (1 - 0.10) * 0.18 ≈ 0.1426

So, the probability that the driver gives a breathalyser test reading that is over the limit is approximately 0.1426.

4. Probability that the driver is under the legal limit, given the breathalyser reading is also below the limit (conditional probability):
P(¬A|¬B) = P(¬A) * P(¬B|¬A) / P(¬B)
P(¬A) = 0.82 (given)
P(¬B|¬A) = 1 - P(B|¬A) = 1 - 0.07 = 0.93
P(¬B) = P(¬B|¬A) * P(¬A) + P(¬B|A) * P(A)
P(¬B) = 0.93 * 0.82 + 0.10 * 0.18 ≈ 0.8954

P(¬A|¬B) = 0.82 * 0.93 / 0.8954 ≈ 0.8539

So, the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit, is approximately 0.8539.

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