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Supposedly a baker mixes 5000 raisins into a large quantity of dough which he mixes well and bakes 1000 equally sized cookies from.

So, I'm going to define a random variable X as number of raisins in a randomly selected cookie. But I'm not sure whether if I should use normal or uniform distribution? X=0 would be a rare occurrence so it would not make sense to use uniform distribution. If I try to use normal distribution, then how would I calculate standard deviation if it isn't given?

Edit: Ok, I left out other part of the problem. I am supposed to find the approximate probability that exactly ten cookies out of 20 randomly selected cookies have each at least four raisins in them.

With this additional information, would it make sense to use hypergeometric distribution instead? X would then be re-defined to the number of cookies out of 20 randomly selected cookies that each contains at least four raisins. Then $P(X=10)$ would be what I want, but it leaves out something about the number of raisins that each cookie contains.

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  • If the raisins are well-mixed into the dough then one can expect them to be roughly uniformly distributed within the dough. – user170231 Mar 14 '18 at 02:50
  • Given a fixed cookie, each raisin has independently probability $p=1/1000$ to be in that cookie. $X$ is then the sum of $n=5000$ independent random variables of that sort (a.k.a Bernoulli with parameter $p$). This means $X$ follows a Binomial distribution with parameters $n,p$. – Clement C. Mar 14 '18 at 02:59
  • We get individual (and intuitively identically distributed) random variables $X_1, X_2, ..., X_{1000}$ that are nonnegative-integer valued and satisfy $\sum_{i=1}^{1000} X_i = 5000$. The actual distribution is not clear and you might model it various ways, one is to say the random variables are constant (each with 5 raisins). Another is to think of it as balls-bins problem and independently and uniformly placing each raisin within each "cookie bin." A "true" distribution likely is tightly centered around 5. – Michael Mar 14 '18 at 02:59
  • @Michael Here, it is for one given cookie (i.e., the different $X$'s for the cookies are not independent, but each of them has a marginal distribution which is Binomial). – Clement C. Mar 14 '18 at 03:00
  • @ClementC. : I gave the situation that can be described, then various models. The Biniomial that you mention is one of the options, but not necessarily the only one. We cannot really know what the distribution will be without more info. I think the "true" distribution would look closer to the "constant 5" than to the independent balls-bins model. (PS: our first two comments are within 5 seconds of each other...which is why I mentioned the balls-bins without commenting that you did hte same thing a few seconds before) – Michael Mar 14 '18 at 03:02
  • @Michael My last comment was just to point out that the OP is asking about the marginals, not really about the dependencies between cookies. – Clement C. Mar 14 '18 at 03:19
  • On the edit and approximation of cookies: Likely you are expeccted to use the independent ball-bin placement model that Clement mentions (leading to a Binomial for $X_1$) and then approximate $X_1, ..., X_{1000}$ as independent Poisson random variables, which is a limiting form of the Binomial random variable involved. – Michael Mar 14 '18 at 04:32

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