an average of 0.56 defects per 500 yards. Using the Poisson formula, find the probability that the number of defects in a given 500-yard piece of this fabric will be more than 3
To calculate the probability using the Poisson formula, we need to know the average number of defects per unit (λ), which in this case is given as 0.56 defects per 500 yards.
The Poisson formula is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
- P(x; λ) is the probability of x events occurring when the average rate is λ
- e is Euler's number, approximately 2.71828
- x is the number of events
- λ is the average rate
In this case, we want to find the probability that the number of defects in a given 500-yard piece of fabric will be more than 3, which means we need to calculate the probability for x > 3.
Let's calculate the probability step by step using the Poisson formula:
1. Find the probability of x = 0, 1, 2, and 3 defects separately:
- For x = 0: P(0; 0.56) = (e^(-0.56) * 0.56^0) / 0! = e^(-0.56) * 1 / 1 = e^(-0.56)
- For x = 1: P(1; 0.56) = (e^(-0.56) * 0.56^1) / 1! = e^(-0.56) * 0.56 / 1 = 0.56 * e^(-0.56)
- For x = 2: P(2; 0.56) = (e^(-0.56) * 0.56^2) / 2! = e^(-0.56) * 0.56^2 / 2 = (0.56^2 / 2) * e^(-0.56)
- For x = 3: P(3; 0.56) = (e^(-0.56) * 0.56^3) / 3! = e^(-0.56) * 0.56^3 / 6 = (0.56^3 / 6) * e^(-0.56)
2. Calculate the probability of having more than 3 defects by subtracting the cumulative probability from 0 (1 minus the sum of the probabilities calculated in step 1):
P(x > 3) = 1 - (P(0; 0.56) + P(1; 0.56) + P(2; 0.56) + P(3; 0.56))
Sum up these values and subtract the result from 1 to get the final probability.
Keep in mind that this calculation assumes the number of defects follows a Poisson distribution and that the average rate remains constant over the 500-yard piece of fabric.