A metal sheet has ,on the average , 5 defects per 1.0 m^2. Assuming a poisson distribution , calculate the probability that a 1.5 m^2 piece of the metal sheet will have at least 4 defects.
To calculate the probability that a 1.5 m^2 piece of the metal sheet will have at least 4 defects, we can use the Poisson distribution formula. Let's break down the steps:
Step 1: Identify the parameters for the Poisson distribution.
In this case, the average number of defects per 1.0 m² is given as 5. This average value is often denoted as λ (lambda) in Poisson distributions.
Step 2: Calculate the average number of defects for a 1.5 m² piece.
To get the average number of defects for a 1.5 m² piece, we can multiply the average number of defects per 1.0 m² by the area of the piece. Therefore, for a 1.5 m² piece, the average number of defects would be 5 * 1.5 = 7.5.
Step 3: Calculate the probability using the Poisson distribution formula.
The formula for the Poisson distribution probability is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
P(x; λ) represents the probability of observing exactly x defects,
λ is the average number of defects per unit area, and
x is the number of defects we want to calculate the probability for.
In this case, we want to calculate the probability of observing at least 4 defects, which means we need to calculate the probability of observing 4, 5, 6, 7, 8, etc. defects. We can calculate this as:
P(x >= 4; 7.5) = 1 - P(x < 4; 7.5)
To calculate P(x < 4; 7.5), we need to sum up the probabilities for x = 0, 1, 2, and 3 using the Poisson distribution formula.
Finally, substitute the values into the formula and calculate the probabilities.
Note: It's helpful to use a calculator or software that offers Poisson distribution calculations, as the calculations can become tedious by hand.