When u leave a party hall, you can do an alcohol test to check whether you are allowed to drive or not. However, the test is not 100% reliable: it only reacts positively on 94% of people who drink too much, and it also responds in 8% on people who do not drink too much. If you know that during this evening 40% of the attendees drank too much, calculate the probability that the test can reacts negatively, if this person has drunk too much. pls help me
To calculate the probability that the test reacts negatively if a person has drunk too much, we need to use conditional probability.
Let's define the following events:
A: Test reacts positively (positive result)
B: Person has drunk too much
A': Test reacts negatively (negative result)
B': Person has not drunk too much
We are given the following information:
P(A|B) = 0.94 (probability of a positive test result given that the person has drunk too much)
P(A'|B') = 0.08 (probability of a negative test result given that the person has not drunk too much)
P(B) = 0.40 (probability that a person has drunk too much)
We need to calculate P(A'|B), which is the probability of a negative test result given that the person has drunk too much.
To find this probability, we can use Bayes' theorem:
P(A'|B) = (P(B') * P(A'|B')) / (P(B) * P(A|B) + P(B') * P(A'|B'))
We need to calculate the values of each component in the formula:
P(B') = 1 - P(B) = 1 - 0.40 = 0.60 (probability that a person has not drunk too much)
P(A|B') = 1 - P(A'|B') = 1 - 0.08 = 0.92 (probability of a positive test result given that the person has not drunk too much)
Now we can substitute these values into Bayes' theorem:
P(A'|B) = (0.60 * 0.92) / (0.40 * 0.94 + 0.60 * 0.08)
Calculating this expression, we find:
P(A'|B) = 0.55
Therefore, the probability that the test reacts negatively if a person has drunk too much is 0.55, or 55%.