If 8% of the population of trees are elm trees, find the probability that in a sample in a sample of 100 trees, there are exactly 6 elm trees. assume the distribution is approximately Poisson
If 8% of the population of trees are elm trees, find the probability that in a sample of 100 trees, there are exactly six elm trees. Assume the distribution is approximately Poisson.
Solution:
Since λ= n(p)= 0.08(100)=8
X=6
(2.7183)^-8 (8)^6 /6!
=0.12213168 (ANSWER)
i need ansower of this question
To find the probability that in a sample of 100 trees, there are exactly 6 elm trees, we can use the Poisson distribution. The Poisson distribution is used to model events that occur randomly over time or space, assuming a constant average rate.
In this case, we can approximate the distribution as Poisson because the number of elm trees in a sample is random and follows a certain rate (8% of the population).
To calculate the probability, we need to determine the average rate (λ) of elm trees in the sample. The Poisson distribution uses the formula:
P(x, λ) = (e^(-λ) * λ^x) / x!
Where x is the number of events we're interested in (6 elm trees in this case), e is the base of natural logarithms (approximately 2.71828), λ is the average rate (in this case, the percentage of elm trees converted to a decimal), and x! is the factorial of x.
First, convert the 8% population proportion to a decimal: 8% = 0.08.
λ = (0.08 * 100) / 100 = 0.08
Now we can substitute the values into the formula:
P(6, 0.08) = (e^(-0.08) * 0.08^6) / 6!
To find the factorial of 6 (6!), we multiply the numbers from 6 down to 1:
6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Now simplify the equation:
P(6, 0.08) = (e^(-0.08) * 0.08^6) / 720
Use a calculator to evaluate e^(-0.08) and 0.08^6:
P(6, 0.08) ≈ (0.9231 * 0.000262144) / 720
P(6, 0.08) ≈ 0.000240778 / 720
P(6, 0.08) ≈ 0.000000334
So, the probability that in a sample of 100 trees there are exactly 6 elm trees, assuming the distribution is approximately Poisson, is approximately 0.000000334.
If 8% of the population of trees are elm trees, find the probability that in a sample of 100 trees, there are exactly six elm trees. Assume the distribution is approximately Poisson.
I need an answer for this question please
Well, let me tell you a joke about trees first:
Why did the tree go to the dentist?
Because it had a cavity!
Now, let's get back to your question. Since we are assuming the distribution to be approximately Poisson, we can use the Poisson formula to calculate the probability. The formula is:
P(x; λ) = (e^(-λ) * λ^x) / x!
In this case, λ (lambda) represents the average number of elm trees in a sample of 100 trees. λ can be calculated as 8% of 100, which equals 8.
So, let's calculate the probability of getting exactly 6 elm trees:
P(6; 8) = (e^(-8) * 8^6) / 6!
Now, let me grab my calculator and do the math for you:
P(6; 8) ≈ 0.103
Therefore, the probability of having exactly 6 elm trees in a sample of 100 trees is approximately 0.103, or 10.3%.