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4. Despite all efforts by the quality control department, the fabric made at Benton Corporation always contains a few defects. A certain type of fabric made at this corporation contains an average of .5 defects per 500 yards. Using the Poisson formula, find the probability that a given piece of 500 yards of this fabric will contain exactly 1 defect.
Ah, fabric with defects, always keeping us on our toes! Alright, let's do some math with a twist of comedy.
To find the probability of exactly 1 defect in 500 yards of fabric made by Benton Corporation, we can use the Poisson formula. Now, get ready for some equation fun!
First, we need to figure out the parameter λ, which represents the average number of defects per unit. In this case, it's given as 0.5 defects per 500 yards. So, 500 yards is our unit.
Now, let's plug in the numbers! We have λ = 0.5 and we want to find the probability of exactly 1 defect, which means x = 1.
P(x=1) = (e^(-λ) * λ^x) / x!
Now, let's simplify this while keeping the fun rolling. Plug in λ = 0.5 and x = 1, and let the magic happen!
P(x=1) = (e^(-0.5) * 0.5^1) / 1!
To make it even more hilarious, let's calculate the values. You'll see, math is all fun and games!
P(x=1) = (0.60653066 * 0.5) / 1
P(x=1) = 0.30326533 / 1
P(x=1) = 0.30326533
And there you have it! The probability of a 500-yard piece of fabric made by Benton Corporation containing exactly 1 defect is approximately 0.3033. Keep in mind, this is no laughing matter when it comes to the seriousness of quality control. But a little humor goes a long way!
To find the probability that a given piece of 500 yards of fabric will contain exactly 1 defect using the Poisson formula, we can use the following equation:
P(x; λ) = (e^-λ * λ^x) / x!
Where:
- P(x; λ) is the probability of having x events occur in a given interval
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average number of events that occur in the interval
- x is the actual number of events that occur in the interval
In this case, we have an average of 0.5 defects per 500 yards of fabric. So, λ = 0.5.
Plugging in these values into the equation, we get:
P(1; 0.5) = (e^-0.5 * 0.5^1) / 1!
Calculating this:
P(1; 0.5) = (e^-0.5 * 0.5) / 1
The value of e^-0.5 is approximately 0.6065.
Substituting this value, we have:
P(1; 0.5) = (0.6065 * 0.5) / 1
P(1; 0.5) = 0.30325
Therefore, the probability of a given piece of 500 yards of fabric containing exactly 1 defect is approximately 0.30325.
To find the probability of a given piece of fabric containing exactly 1 defect, we will use the Poisson formula.
The Poisson formula, used in probability theory, calculates the probability of a certain number of events occurring in a fixed interval of time or space, given the average rate of occurrence. In this case, the average rate of defects is given as 0.5 defects per 500 yards.
The formula is as follows:
P(x; μ) = (e^(-μ) * μ^x) / x!
Where:
- P(x; μ) is the probability of x events occurring, given the average rate μ.
- e is the base of the natural logarithm (approximately 2.71828).
- μ is the average rate of occurrence.
- x is the number of events we are interested in.
- x! denotes the factorial of x.
In our case, we want to find the probability of a fabric piece containing exactly 1 defect, so x = 1. The average rate of defects is 0.5 defects per 500 yards, which means μ = 0.5.
Let's substitute the values into the formula and calculate the probability:
P(1; 0.5) = (e^(-0.5) * 0.5^1) / 1!
Calculating e^(-0.5) gives us approximately 0.60653, and 0.5^1 is simply 0.5. The factorial of 1 is 1.
Substituting these values into the formula:
P(1; 0.5) = (0.60653 * 0.5) / 1
P(1; 0.5) = 0.30326
Therefore, the probability that a given piece of 500 yards of fabric will contain exactly 1 defect is approximately 0.30326, or 30.33%.