Suppose that the number of accidents occurring in an industrial plant is described by a Poisson process with an average of 1.5 accidents every three months. During the last three months, four accidents occurred.
(a)probability that no more than 12 accidents will occur during a particular year.
(b)probability that no accidents will occur during a particular year.
the mean is based off of 3 months. multiply it out to get the mean for a correct time period (1.5x4=6). I use the Poisson function in excel. Mean is equal to 6 and x is equal to 12 and 0. True is up to a certain number (up to 12) and false is exactly a certain number (exactly 0). In this case the answers are .9912 and .0025. Good luck
To solve these probabilities, we will use the properties of a Poisson distribution.
The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence.
The average rate of occurrence is given as 1.5 accidents every three months. To find the average rate per year, we need to convert three months to a year.
There are 12 months in a year, so the average rate per year is (1.5 accidents / 3 months) * 12 months = 6 accidents per year.
Let's calculate the probabilities:
(a) probability that no more than 12 accidents will occur during a particular year.
To calculate this probability, we need to sum the probabilities of having 0, 1, 2, ..., 12 accidents.
P(X ≤ 12) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 12)
Where X is the number of accidents occurring in a particular year, and P(X = k) represents the probability of having exactly k accidents.
Using the Poisson probability formula, the probability of having k accidents in a year is calculated as:
P(X = k) = (e^(-λ) * λ^k) / k!
Where e is the base of the natural logarithm (approximately 2.71828), λ is the average number of accidents per year, and k is the number of accidents we want to calculate the probability for.
So, for each k from 0 to 12, we can substitute these values into the formula and calculate the probabilities.
(b) probability that no accidents will occur during a particular year.
To calculate this probability, we need to find P(X = 0).
Using the same formula mentioned above, we can substitute k = 0 and calculate the probability.
Let's carry out the calculations:
(a) The probability that no more than 12 accidents will occur during a particular year:
P(X ≤ 12) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 12)
We will calculate the probabilities for each k from 0 to 12 and sum them up.
(b) The probability that no accidents will occur during a particular year:
P(X = 0)
By substituting the respective values into the Poisson probability formula, we will get the exact values for the probabilities.