# Difficult Poisson Variable

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• Dec 9th 2008, 07:46 PM
bondvista
Difficult Poisson Variable
Hi everyone
I have encountered a difficult problems on Poisson Variable. I hope you guys can give me some help. I guess I did part a) right but I really don't know how to do b). Thank you in advance.

Here it is:
Amy is trying out a new recipe for raisin bread. Each batch of bread dough makes 3 loaves. And each loaf contains 20 slices of bread.

a) If she puts 100 raisins into a batch of dough, what is the probability that a randomly selected slice of bread contains no raisins?

My answer for a) is: Let X = # of raisins in slice of bread.
X is a Poisson variable.
The events = # of raisins in the bread "integral" = A slice of the bread.
Mean Lambda = 100/(3*20) = 1.667 = 5/3
Want probability(no Raisins)
P(X=0) = (5/3)^0 * e^(-5/3) / 0! = 0.18887

b) is the difficult one
How many raisins Amy must put into a batch of dough so that the probability that a randomly selected slice of bread will have no raisins is 0.01??
• Dec 9th 2008, 07:59 PM
meymathis
I don't see how the solution to a) could be correct. A Poisson variable has infinite support. In other words there is a finite probability that a slice might have 101 (or 10000001) raisins in it even if it is small.

So you have 100 raisins and 60 slices. What is the probability that a randomly selected slice of bread has not raisins. Ok, pick a slice. Each raisin has a 1/60 probability of being in that slice. Let $\displaystyle X$ be the number of raisins in that slice. Then $\displaystyle X$ has a binomial distribution with probability of success $\displaystyle p=1/60$. The question asks what is

$\displaystyle \mathrm{P}\left(X=0\right)$

Can you finish this part off?

If you get a) now, b should be straight forward.
• Dec 9th 2008, 08:08 PM
bondvista
Thank you for supporting
Hi there
Amy only got 100 raisins though. I don't understand why you got 101 or 1000001?? My class is lower division. The thing you are talking about is probably upper division.

Anyways, I don't understand the way you did it. Would you please completely solve the problem so that I can see how you do it your way. Thank you!
• Dec 9th 2008, 08:35 PM
meymathis
What I was trying to say was that a Poisson random variable does not fit the problem. Something that has a Poisson distribution has infinitely many outcomes, any number 0 and up (including 101 and 10000001). See Poisson distribution - Wikipedia, the free encyclopedia

If X is defined as the number of raisins in a slice of bread, then there are 101 total outcomes (any number from 0 to 100). So X cannot be Poisson. X as it turns out has a binomial distribution. See Binomial distribution - Wikipedia, the free encyclopedia.

A particular raisin being in your slice of bread is one "trial", with probability of success 1/60 (p). There are 100 trials (n).

From the web page, you can see that
$\displaystyle \mathrm{P}(X=k) = {n \choose k}p^k(1-p)^{n-k}$

You want to calculate $\displaystyle \mathrm{P}(X=0)$.

I prefer to not do the entire problem out for people in most cases. You learn the best by figuring stuff out on your own, in general. So I am trying to lead you through it.
• Dec 10th 2008, 04:26 PM
awkward
I can't resist interjecting here that the Poisson may indeed be a good choice of distribution here, despite the objections raised by meymathis. In fact, the Wikipedia page on the Poisson distribution (Poisson distribution - Wikipedia, the free encyclopedia) cites a rule of thumb that the Poisson is an excellent approximation to the Binomial distribution if $\displaystyle n \geq 100$ and $\displaystyle np \leq 10$. Here we have $\displaystyle n = 100$ and $\displaystyle np = 1.667$, so both criteria are met.

If you "do the math" I think you will find that the answer to a) using the Poisson approximation is 0.1889 (as found by bondvista), and the answer using the Binomial distribution is 0.1862-- pretty close.
• Dec 11th 2008, 08:00 AM
meymathis
Quote:

Originally Posted by awkward
I can't resist interjecting here that the Poisson may indeed be a good choice of distribution here, despite the objections raised by meymathis. In fact, the Wikipedia page on the Poisson distribution (Poisson distribution - Wikipedia, the free encyclopedia) cites a rule of thumb that the Poisson is an excellent approximation to the Binomial distribution if $\displaystyle n \geq 100$ and $\displaystyle np \leq 10$. Here we have $\displaystyle n = 100$ and $\displaystyle np = 1.667$, so both criteria are met.

If you "do the math" I think you will find that the answer to a) using the Poisson approximation is 0.1889 (as found by bondvista), and the answer using the Binomial distribution is 0.1862-- pretty close.

Thanks for pointing that out. I would still argue that while it may be a good approximation, there does not appear to be any reason to approximate it. Why use an approximation unless it either makes your life easier or you are instructed to do so? Perhaps the OP was instructed to, but it wasn't made clear that the problem required the use of Poisson (though perhaps that was what was meant by I have encountered a difficult problems on Poisson Variable).

Quote:

Originally Posted by bondvista
Hi everyone
I have encountered a difficult problems on Poisson Variable. I hope you guys can give me some help. I guess I did part a) right but I really don't know how to do b). Thank you in advance.

Here it is:
Amy is trying out a new recipe for raisin bread. Each batch of bread dough makes 3 loaves. And each loaf contains 20 slices of bread.

a) If she puts 100 raisins into a batch of dough, what is the probability that a randomly selected slice of bread contains no raisins?

My answer for a) is: Let X = # of raisins in slice of bread.
X is a Poisson variable.
The events = # of raisins in the bread "integral" = A slice of the bread.
Mean Lambda = 100/(3*20) = 1.667 = 5/3
Want probability(no Raisins)
P(X=0) = (5/3)^0 * e^(-5/3) / 0! = 0.18887

If you were supposed to use Poisson approximation then what you did is correct. Sorry for the confusion.

Quote:

b) is the difficult one
How many raisins Amy must put into a batch of dough so that the probability that a randomly selected slice of bread will have no raisins is 0.01??

If we just do what you did but leave the number of raisins variable, which we will call $\displaystyle n$. Then

$\displaystyle \lambda = n/(3\cdot 20)=n/60$

$\displaystyle \mathbf{P}(X=0) = \lambda^0 \cdot \frac{e^{-\lambda}}{0!} = e^{-n/60}$

Now the question asks for what value of $\displaystyle n$ will make $\displaystyle \mathbf{P}(X=0) = 0.01$

$\displaystyle 0.01 = e^{-n/60}$

Now just solve for $\displaystyle n$.

For the record, you get slightly different answers (as you might expect) if you use the Poisson approximation versus the Binomial distribution. One you get 274 and the other 276. (Nerd)