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 solve this problem, we can use the Poisson distribution formula.

The formula for the Poisson distribution is:

P(X = x) = (e^(-λ) * λ^x) / x!

Where:
- P(X = x) is the probability of having exactly x events occur within a given area or time period.
- λ is the average number of events occurring in the given area or time period.
- e is a mathematical constant approximately equal to 2.71828.
- x is the number of events we are interested in.

In this case, the average number of defects per 1.0 m^2 is given as 5, so λ = 5.

To calculate the probability that a 1.5 m^2 piece of the metal sheet will have at least 4 defects, we need to sum the probabilities of getting 4, 5, 6, 7, ..., up to infinity defects.

P(at least 4 defects) = P(4 defects) + P(5 defects) + P(6 defects) + ...

We can calculate each term using the Poisson distribution formula and then sum them up.

Let's calculate the probability step-by-step:

Step 1: Calculate the probability of exactly x defects using the Poisson distribution formula.
P(X = x) = (e^(-λ) * λ^x) / x!

Step 2: Calculate the probability of at least 4 defects.
P(at least 4 defects) = P(4 defects) + P(5 defects) + P(6 defects) + ...

For this calculation, we will use x = 4, 5, 6, 7, ...

Step 3: Sum up the probabilities calculated in step 2 to get the final probability.

Let's calculate the probabilities for each x value and then sum them up:

P(X = 4) = (e^(-5) * 5^4) / 4!
P(X = 5) = (e^(-5) * 5^5) / 5!
P(X = 6) = (e^(-5) * 5^6) / 6!
P(X = 7) = (e^(-5) * 5^7) / 7!
...

We'll calculate each term and sum them up to get the final probability.

To solve this problem, we need to use the Poisson distribution formula. The Poisson distribution is used to model the probability of a certain number of events occurring within a fixed interval of time or space when these events occur at a constant average rate.

The formula for the Poisson distribution is:

P(x; λ) = (e^(-λ) * λ^x) / x!

Where:
P(x; λ) is the probability of x events occurring,
e is the base of the natural logarithm (approximately 2.71828),
λ (lambda) is the average number of events per interval,
x is the actual number of events occurring, and
x! represents the factorial of x.

In this problem, the average number of defects per 1.0 m^2 is given as 5. So λ (lambda) is 5.

To find the probability that a 1.5 m^2 piece of the metal sheet will have at least 4 defects, we need to calculate the probability of having 4, 5, 6, 7, and so on defects, up to infinity. Then we sum up these individual probabilities.

Let's break down the calculation into steps:

Step 1: Calculate the probability of having exactly 4 defects in a 1.5 m^2 piece.
Using the Poisson distribution formula:
P(4; 5) = (e^(-5) * 5^4) / 4!

Step 2: Calculate the probability of having exactly 5 defects in a 1.5 m^2 piece.
P(5; 5) = (e^(-5) * 5^5) / 5!

Step 3: Calculate the probability of having exactly 6 defects in a 1.5 m^2 piece.
P(6; 5) = (e^(-5) * 5^6) / 6!

Step 4: Continue this process for 7 defects, 8 defects, and so on until you reach a reasonable number of defects to approximate "at least 4".

Step 5: Sum up all the probabilities obtained from Step 1 to Step 4.

By following these steps, you can calculate the probability that a 1.5 m^2 piece of the metal sheet will have at least 4 defects using the Poisson distribution.