The number of traffic accidents per day on a certain section of highway in the past was 4 accidents per day in average. Based on this,
a) What is the probability that there may be exactly two traffic accidents tomorrow?
b) What is the probability that there may be no traffic accident tomorrow?
c) What is the probability that there may be more than five traffic accidents tomorrow?
d) What is the probability that there may be no more than two traffic accidents tomorrow?
To answer these probability questions, we will use the concept of the Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space.
The Poisson distribution requires one parameter, which is the average rate or mean value of the events. In this case, the average rate is given as 4 accidents per day.
a) To calculate the probability of exactly two traffic accidents tomorrow, we can use the formula for the Poisson distribution:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
X is the random variable representing the number of accidents
k is the specific number of accidents we want to calculate the probability for
λ is the average rate of accidents per day
Using the given average rate of 4 accidents per day and the value of k as 2, we can substitute these values into the formula and calculate the probability:
P(X = 2) = (e^(-4) * 4^2) / 2!
b) To calculate the probability of no traffic accidents tomorrow, we need to find the probability when k is 0. Using the same formula as above, we substitute k = 0:
P(X = 0) = (e^(-4) * 4^0) / 0!
c) To calculate the probability of more than five traffic accidents tomorrow, we need to find the sum of probabilities for k = 6, 7, 8, and so on until infinity:
P(X > 5) = 1 - (P(X ≤ 5))
To calculate P(X ≤ 5), we can find the sum of probabilities for k = 0, 1, 2, 3, 4, 5.
P(X > 5) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)]
d) To calculate the probability of no more than two traffic accidents tomorrow, we need to find the sum of probabilities for k = 0, 1, and 2:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)