Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared centimeters of dust contains more than 10090 particles?
Round your answer to 3 decimal places.
To solve this problem, we can use the Poisson probability distribution formula. The formula for the Poisson distribution is given by:
P(X = x) = (e^(-λ) * λ^x) / x!
Where:
- P(X = x) is the probability of observing exactly x events
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average rate or mean of the events (in this case, 1000)
- x is the number of events we are interested in
In this case, we want to find the probability that 10 squared centimeters of dust contains more than 10,090 particles. This can be expressed as:
P(X > 10090) = 1 - P(X ≤ 10090)
To find the probability that X is less than or equal to 10090, we need to sum up the individual probabilities for all values from 0 to 10090 using the Poisson distribution formula. However, calculating this directly would be computationally intensive and time-consuming.
Fortunately, for large values of λ (in this case, 1000), the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ. With this approximation, we can use the cumulative distribution function (CDF) of the normal distribution to calculate P(X ≤ 10090).
The formula for the CDF of the normal distribution is:
P(X ≤ x) = Φ((x - μ) / σ)
Where:
- Φ is the standard normal CDF
- μ is the mean of the normal distribution (λ in this case)
- σ is the standard deviation of the normal distribution (√λ)
In our case, μ = 1000 and σ = √1000 = 31.62.
Now, we can calculate P(X ≤ 10090) using a standard normal distribution table or a calculator that provides the CDF value. The calculation would be:
P(X ≤ 10090) = Φ((10090 - 1000) / 31.62)
After finding this value, we can subtract it from 1 to get the probability of having more than 10,090 particles:
P(X > 10090) = 1 - P(X ≤ 10090)
Finally, you can round the result to 3 decimal places to get the answer.