The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.028 failure per hour.
(a) What is the probability that the instrument does not fail in an 8-hour shift?
(b) What is the probability of at least one failure in a 24-hour day?
To solve this problem, we will use the Poisson probability formula. The formula for the probability mass function (PMF) of a Poisson random variable is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
where X is the random variable, λ is the average number of events per interval, k is the number of events we are interested in, and e is Euler's number (approximately 2.71828).
(a) Probability that the instrument does not fail in an 8-hour shift:
For this scenario, we need to find the probability that X = 0, where X represents the number of failures in an 8-hour shift. The average number of failures in 1 hour is given as 0.028, so the average number of failures in an 8-hour shift would be 8 * 0.028 = 0.224.
Using the Poisson probability formula, we can calculate:
P(X = 0) = (e^(-0.224) * 0.224^0) / 0!
Since any number raised to the power of 0 is 1, and 0! is also 1, the formula simplifies to:
P(X = 0) = e^(-0.224)
To find the numerical value of e^(-0.224), you can use a scientific calculator or computer software that has a function for calculating exponentials.
(b) Probability of at least one failure in a 24-hour day:
For this scenario, we need to find the probability that X ≥ 1. Using the complement rule, we can calculate the probability of no failures and subtract it from 1.
P(X ≥ 1) = 1 - P(X = 0)
We already know from part (a) that P(X = 0) is e^(-0.224), so we can substitute that value into the formula:
P(X ≥ 1) = 1 - e^(-0.224)
Again, you can use a calculator or computer software to find the numerical value of e^(-0.224).
By calculating both probabilities, you will have the answers to both parts of the question.
To solve these problems, we will use the Poisson distribution formula:
P(x; λ) = (e^(-λ)*λ^x) / x!
Where:
- P(x; λ) is the probability of having x events
- e is the mathematical constant approximately equal to 2.71828
- λ is the average number of events per interval
- x is the number of events
(a) Probability that the instrument does not fail in an 8-hour shift:
In this case, λ = 0.028 failures per hour, and we want to calculate the probability of having zero failures in an 8-hour shift.
To do this, we substitute λ = 0.028 and x = 0 into the Poisson distribution formula:
P(x=0; λ=0.028) = (e^(-0.028)*0.028^0) / 0!
= (e^(-0.028)*1) / 1
= e^(-0.028)
≈ 0.972
Therefore, the probability that the instrument does not fail in an 8-hour shift is approximately 0.972, or 97.2%.
(b) Probability of at least one failure in a 24-hour day:
In this case, we again have λ = 0.028 failures per hour, and we want to calculate the probability of having at least one failure in a 24-hour day.
Since we want the probability of having at least one failure, we can calculate the complement probability of having zero failures and subtract it from 1.
The complement probability can be calculated using the formula:
P(x=0; λ=0.028) = (e^(-0.028)*0.028^0) / 0!
= (e^(-0.028)*1) / 1
= e^(-0.028)
≈ 0.972
Therefore, the probability of having at least one failure in a 24-hour day is approximately 1 - 0.972 = 0.028, or 2.8%.
Poisson distribution (m = mean):
P(x) = e^(-m) m^x / x!
For (a):
Take (0.028)(8) = 0.224 (this is the mean). Find P(0) using the above formula.
For (b):
Take (0.028)(24) = 0.672 (this is the mean). Find P(0) using the above formula. Then subtract that value from 1. This will be your probability.
I hope this will help get you started.