During a study of auto accidents, the highway safety council found that 60% of all accidents 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 occured at night given that it was alcohol related.
It would help if you proofread your questions before you posted theme.e., "60% of all accidents occur at night 52% are alcohol related and 37% occur at night and are alcohol related".
Oh, we're going in-depth with statistics now? Hold on, let me put on my geeky glasses. *Puts on oversized novelty glasses*
(a) Well, to find the probability that an accident was alcohol related given that it occurred at night, we need to use conditional probability. So, we take the percentage of accidents that occur at night and are alcohol related (37%), and divide it by the percentage of accidents that occur at night (60%).
So, the probability that an accident was alcohol related given that it occurred at night is 37% / 60%, which simplifies to approximately 0.6167. Or, in other words, about a 61.67% chance of the accident being alcohol related if it happened at night.
(b) Now, let's calculate the probability that an accident occurred at night given that it was alcohol related. We'll use the same principle of conditional probability again. This time, we take the percentage of accidents that occur at night and are alcohol related (37%) and divide it by the percentage of accidents that are alcohol related (52%).
So, the probability that an accident occurred at night given that it was alcohol related is 37% / 52%, which simplifies to around 0.7115. In plain terms, there's roughly a 71.15% chance that an accident occurred at night if it was alcohol related.
Remember, statistics can be a bit tricky, but humor always helps lighten things up!
To solve this problem, we can use conditional probability.
Let's define the events:
A = Accident is alcohol related
N = Accident occurs at night
Given information:
P(N) = 0.60 (probability of an accident occurring at night)
P(A) = 0.52 (probability of an accident being alcohol related)
P(N ∩ A) = 0.37 (probability of an accident occurring at night and being alcohol related)
Now, let's calculate the probabilities for the given conditions:
(a) What is the probability that an accident was alcohol related given that it occurred at night?
To find P(A|N), we use the formula: P(A|N) = P(N ∩ A) / P(N)
P(A|N) = 0.37 / 0.60
P(A|N) = 0.617
Therefore, the probability that an accident was alcohol-related given that it occurred at night is 0.617 or 61.7%.
(b) What is the probability that an accident occurred at night given that it was alcohol related?
To find P(N|A), we use the formula: P(N|A) = P(N ∩ A) / P(A)
P(N|A) = 0.37 / 0.52
P(N|A) = 0.712
Therefore, the probability that an accident occurred at night given that it was alcohol related is 0.712 or 71.2%.
To find the probability of events happening given another event, we can use conditional probability. Let's calculate the probability for both parts:
(a) We need to find the probability that an accident was alcohol related, given that it occurred at night. This can be denoted as P(A|N), where A represents "accident is alcohol related" and N represents "accident occurs at night."
We are given:
P(A) = 0.52 (probability of an accident being alcohol related)
P(N) = 0.60 (probability of an accident occurring at night)
P(A∩N) = 0.37 (probability that an accident is both alcohol related and occurs at night)
We can use the conditional probability formula:
P(A|N) = P(A∩N) / P(N)
Substituting the values:
P(A|N) = 0.37 / 0.60
P(A|N) ≈ 0.6167
Therefore, the probability that an accident was alcohol related given that it occurred at night is approximately 0.6167 or 61.67%.
(b) Now, we need to find the probability that an accident occurred at night, given that it was alcohol related. This can be denoted as P(N|A).
Using Bayes' theorem, we have:
P(N|A) = (P(A|N) * P(N)) / P(A)
We already calculated P(A|N) as approximately 0.6167, P(N) is still 0.60, and P(A) is given as 0.52.
Substituting the values:
P(N|A) = (0.6167 * 0.60) / 0.52
P(N|A) ≈ 0.7115
Therefore, the probability that an accident occurred at night given that it was alcohol related is approximately 0.7115 or 71.15%.
N = probability of accidents that happens at night
A = " " " that are alcohol related
P(N) = 0.6, P(A) = 0.52 and P(AnN) = 0.37
P(A/N) = P(AnN)/p(N) = 0.37/ 0.52 = 0.71
P(N/A) = P(AnN)/p(A) = 0.37/0.6 = 0.61