During a study of auto accident,the highway safety council found that 60%of all accident occur at night,52%are alcohol-related,and 37% occur at night and are alcohol related :a)What is the probability that an accident was alcohol-related,given that it occurred at night ?b)What is the probability that an accident occurred at night,given that it was alcohol-related ?
(a) 37% / 60%
(b) 37% / 52%
To solve these probability problems, we can use conditional probability. Let's use the following notation:
A = Accident occurred at night
B = Accident was alcohol-related
a) To find the probability that an accident was alcohol-related given that it occurred at night, we can use the formula for conditional probability:
P(B|A) = P(A ∩ B) / P(A)
We are given that the probability of an accident occurring at night (P(A)) is 0.60, the probability of an accident being alcohol-related (P(B)) is 0.52, and the probability of an accident occurring at night and being alcohol-related (P(A ∩ B)) is 0.37.
Substituting these values into the formula:
P(B|A) = 0.37 / 0.60 ≈ 0.617
Therefore, the probability that an accident was alcohol-related, given that it occurred at night, is approximately 0.617.
b) To find the probability that an accident occurred at night, given that it was alcohol-related, we can again use the formula for conditional probability:
P(A|B) = P(A ∩ B) / P(B)
Using the given values:
P(A) = 0.60
P(B) = 0.52
P(A ∩ B) = 0.37
Substituting these values into the formula:
P(A|B) = 0.37 / 0.52 ≈ 0.712
Therefore, the probability that an accident occurred at night, given that it was alcohol-related, is approximately 0.712.
To find the answers to these probability questions, we can use conditional probability. Conditional probability is the probability of event A happening given that event B has already occurred. In this case, event A is an accident being alcohol-related, and event B is an accident occurring at night.
a) What is the probability that an accident was alcohol-related, given that it occurred at night?
This can be calculated using the formula for conditional probability:
P(A|B) = P(A and B) / P(B)
In this case, P(A and B) refers to the probability that an accident occurred at night and was alcohol-related, and P(B) refers to the probability that an accident occurred at night.
The probability of an accident occurring at night and being alcohol-related is given as 37%, or 0.37. The probability of an accident occurring at night is given as 60%, or 0.60.
P(A|B) = 0.37 / 0.60 ≈ 0.617
Therefore, the probability that an accident was alcohol-related, given that it occurred at night, is approximately 0.617 or 61.7%.
b) What is the probability that an accident occurred at night, given that it was alcohol-related?
This can also be calculated using the conditional probability formula:
P(B|A) = P(A and B) / P(A)
In this case, P(A and B) refers to the probability that an accident occurred at night and was alcohol-related, and P(A) refers to the probability that an accident was alcohol-related.
We already know that the probability of an accident occurring at night and being alcohol-related is 0.37. The probability of an accident being alcohol-related is given as 52%, or 0.52.
P(B|A) = 0.37 / 0.52 ≈ 0.712
Therefore, the probability that an accident occurred at night, given that it was alcohol-related, is approximately 0.712 or 71.2%.