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: 8 % 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 15 % 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 following?
To answer the questions, we will use conditional probabilities. Let's assign the following notations:
A = the event that a randomly stopped driver is above the legal alcohol limit.
B = the event that the breathalyser test indicates a reading above the legal limit.
Now, we can calculate the requested probabilities step-by-step.
1. Find the probability that a driver gives a reading above the legal limit, even though they have not consumed an excess of alcohol (P(B|A')):
Given that "8% of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit," we can deduce that P(B|A') = 0.08.
2. Find the probability that a driver gives a reading below the legal limit, even though they are above the legal alcohol limit (P(B'|A)):
Given that "10% of drivers who are above the legal limit will give a reading below that level," we can deduce that P(B'|A) = 0.10.
3. Find the probability that a randomly stopped driver is above the legal alcohol limit (P(A)):
Given that "in fact, 15% of drivers are above the legal alcohol limit," we can deduce that P(A) = 0.15.
4. Find the probability that a randomly stopped driver is below the legal alcohol limit (P(A')):
Since the total probability must sum to 1, we can calculate P(A') as 1 - P(A):
P(A') = 1 - P(A) = 1 - 0.15 = 0.85.
Now, using the given information, we can calculate the remaining probabilities step-by-step.
5. Find the probability that a driver gives a reading above the legal limit (P(B)):
Using the law of total probability, we can calculate P(B) as:
P(B) = P(B|A) * P(A) + P(B|A') * P(A').
P(B) = 0.10 * 0.15 + 0.08 * 0.85 ≈ 0.019 + 0.068 ≈ 0.087.
Therefore, P(B) ≈ 0.087.
6. Find the probability that a randomly stopped driver is above the legal alcohol limit, given that the breathalyser test indicates a reading above the legal limit (P(A|B)):
Using Bayes' theorem, we can calculate P(A|B) as:
P(A|B) = (P(B|A) * P(A)) / P(B).
P(A|B) = (0.10 * 0.15) / 0.087 ≈ 0.015 / 0.087 ≈ 0.172.
Therefore, P(A|B) ≈ 0.172.
7. Find the probability that a randomly stopped driver is below the legal alcohol limit, given that the breathalyser test indicates a reading below the legal limit (P(A'|B')):
Using Bayes' theorem, we can calculate P(A'|B') as:
P(A'|B') = (P(B'|A') * P(A')) / P(B').
P(A'|B') = (0.08 * 0.85) / (1 - P(B)).
P(A'|B') = (0.08 * 0.85) / (1 - 0.087) ≈ 0.068 / 0.913 ≈ 0.074.
Therefore, P(A'|B') ≈ 0.074.
These are the answers to the questions, rounded to four decimal places:
- P(B) ≈ 0.087
- P(A|B) ≈ 0.172
- P(A'|B') ≈ 0.074
To find the answers to the questions, we will use conditional probability, specifically Bayes' theorem.
Let's define the following events:
A = the driver has consumed an excess of alcohol
B = the breathalyser test gives a reading above the legal limit
We are given the following probabilities:
P(B|not A) = 0.08 (probability of getting a positive test result when the driver has not consumed an excess of alcohol)
P(not B|A) = 0.10 (probability of getting a negative test result when the driver has consumed an excess of alcohol)
P(A) = 0.15 (probability of a driver having consumed an excess of alcohol)
Now, let's calculate the required probabilities one by one.
1. What is the probability that a driver has consumed an excess of alcohol and the breathalyser test gives a reading above the legal limit? (P(A and B))
Using Bayes' theorem:
P(A and B) = P(B|A) * P(A) = 0.90 * 0.15 = 0.135
2. What is the probability that the breathalyser test gives a reading above the legal limit? (P(B))
To find this, we need to consider two scenarios: when the driver has consumed an excess of alcohol (A) and when the driver has not consumed an excess of alcohol (not A).
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
= 0.90 * 0.15 + 0.08 * (1 - 0.15)
= 0.135 + 0.068
= 0.203
3. What is the probability that a driver has consumed an excess of alcohol given that the breathalyser test gives a reading above the legal limit? (P(A|B))
Using Bayes' theorem:
P(A|B) = P(B|A) * P(A) / P(B)
= 0.90 * 0.15 / 0.203
= 0.135 / 0.203
≈ 0.6640
4. What is the probability that the breathalyser test gives a reading above the legal limit given that the driver has not consumed an excess of alcohol? (P(B|not A))
This is given as 0.08.
So, to four decimal places, we have:
1. P(A and B) ≈ 0.135
2. P(B) ≈ 0.203
3. P(A|B) ≈ 0.6640
4. P(B|not A) = 0.080