Poisson distribution question

In summary, The probability that exactly 3 of the next 20,000 babies born will have the mutation is calculated using the Poisson distribution with p = 1/10,000 and n = 20,000. The formula used is (e^(-np) * (np)^k / k!) and the final answer is calculated for k = 3. This is because we are looking for exactly three occurrences, not at most three.
  • #1
semidevil
157
2
ok, so on average, there is a chromosome mutation link once every 10,000 baby births.

approximate the probability that exactly 3 of the next 20,000 babies born will have the mutation.

so using poisson distribution, I let
p = 1/10,000
n = 20,000.

and use formula (e^(-np) * (np)^k / k!

so when I do the calculation, do I sum it all up for k = 1, 2, 3, or do I just calculate k = 3 and that is my final answer?

if they ask at most 3, then I would sum up k = 1, 2, 3 right?

I first applied the formula for k = 1, 2, 3, but then it got me thinking about it...
 
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  • #2
The answer is just for k = 3. You are looking for exactly three of the next 20,000 babies born to have the mutation, not at most three. Therefore, you only need to calculate the probability for k = 3.
 
  • #3



Your approach is correct. You would use the Poisson distribution formula to calculate the probability of exactly 3 babies out of 20,000 having the mutation. This would involve calculating the probability for k = 3 only.

If the question asked for at most 3 babies, then you would need to sum up the probabilities for k = 1, 2, and 3. This would give you the probability of 3 or fewer babies having the mutation.

It is important to carefully read and understand the question to determine which approach to use. In this case, the question specifically asks for the probability of exactly 3 babies having the mutation, so you would only need to calculate for k = 3.
 

What is a Poisson distribution?

A Poisson distribution is a probability distribution that is used to describe the number of occurrences of a certain event within a specific time or space interval. It is typically used when the events are independent and occur at a constant rate.

What are the key characteristics of a Poisson distribution?

The key characteristics of a Poisson distribution include: being a discrete distribution, having a mean and variance that are equal, and being right-skewed (or positively skewed).

How is a Poisson distribution different from a normal distribution?

While both a Poisson distribution and a normal distribution are used to describe probability, they differ in several ways. A normal distribution is continuous and symmetrical, while a Poisson distribution is discrete and right-skewed. Additionally, a normal distribution can take on any value, while a Poisson distribution is limited to non-negative integers.

What types of real-life phenomena can be modeled using a Poisson distribution?

A Poisson distribution can be used to model a variety of real-life phenomena, such as the number of customers arriving at a store in a given hour, the number of accidents on a highway in a day, or the number of defects in a batch of products. Essentially, any situation where events occur at a constant rate and are independent of each other can be modeled using a Poisson distribution.

How is a Poisson distribution calculated?

The Poisson distribution can be calculated using the formula P(x) = (e^-λ * λ^x) / x!, where x is the number of events, e is the mathematical constant, and λ is the average rate of events. This formula allows us to calculate the probability of a specific number of events occurring within a given time or space interval.

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