# Poisson Distribution - Selecting cookies that are indistinguishable

• B
• domingoleung
In summary, the conversation was about a chef making cookies with nuts and the probability of getting certain types of cookies. For part (a), there are two ways to calculate the probability of getting a specific type of cookie as the fifth one chosen. For part (b), there are two approaches to calculating the probability of getting a box with at most two cookies without any nuts.
domingoleung
TL;DR Summary
Cookies that might have hazelnuts or almonds.
Here's the problem:
A chef made 500 cookies randomly mixed with 1000 nuts including 600 almonds and 400 hazelnuts in which each nut is the same size. Suppose the number of pieces of nuts in a piece of cookie follows a Poisson distribution.
(a) Suppose cookies are randomly selected one-by-one with replacement, find the probability that the fifth chosen cookie will be the 1st cookie without any hazelnut?
(b) Suppose 12 cookies are packed into each box. What is the probability of getting a randomly selected box with at most two piecs of cookies without any nuts?

My Appraoch:
For (a), I have two two ways of doing so:
1.
λ hazelnuts = 400/500 = 0.8
= (1- e-0.8 (0.8)0 )4 * e-0.8 (0.8)0
= 0.0413
2.
λ hazelnuts = 400/500 = 0.8
P(1st cookie without hazelnut) x 1/500
= e-0.8 (0.8)0 x 1/500
= 8.98 x 10-4

For (b),
First Approach:
λ nuts = 1000/500 = 2
P(no nuts)
= e-2 (2)0
= 0.1353
λ no nuts in 12 cookies = 12 x 0.1353 = 1.624
P(At most 2 cookies without any nuts)
= e-1.624 (1.624)0 + e-1.624 (1.624)1 + e-1.624 (1.624)2 / 2!
= 0.7771

Second Approach:
λ nuts = 1000/500 = 2
P( 1 cookie without any nuts and 11 cookies with nuts) + P( 2 cookies with no nuts and 10 cookies with nuts ) + P(All cookies with nuts)
= (1 - e-2 (2)0)11 x e-2 (2)0 + 10C2 * (1 - e-2 (2)0)10 x (e-2 (2)0)2 + (1 - e-2 (2)0)12
= 0.3945
To me, the two approaches for each sub question just make sense to me. Please tell me if I did something wrong :/

HI,

re a) 2: what are you calcluating there ?

Re b) 1:
domingoleung said:
λ no nuts in 12 cookies = 12 x 0.1353 = 1.624
What's this ?

## 1. What is Poisson distribution?

Poisson distribution is a statistical distribution that describes the probability of a certain number of events occurring within a specific time or space, given the average rate of occurrence and the independence of events.

## 2. How is Poisson distribution used in selecting indistinguishable cookies?

Poisson distribution can be used to calculate the probability of selecting a certain number of indistinguishable cookies from a larger batch. This is because the distribution assumes that each cookie is equally likely to be selected and that the selection of one cookie does not affect the selection of another.

## 3. What factors affect the accuracy of using Poisson distribution for selecting cookies?

The accuracy of using Poisson distribution for selecting cookies depends on the assumption that the average rate of cookie selection remains constant and that the selection of one cookie does not affect the selection of another. If these assumptions are not met, the accuracy of the distribution may be affected.

## 4. How can Poisson distribution be applied to real-world scenarios?

Poisson distribution can be applied to various real-world scenarios, such as predicting the number of accidents on a road in a given time period, estimating the number of customers entering a store in an hour, or determining the probability of a certain number of defects in a batch of products.

## 5. Are there any limitations to using Poisson distribution for selecting cookies?

One limitation of using Poisson distribution for selecting cookies is that it assumes a constant average rate of cookie selection, which may not always be the case in real-world scenarios. Additionally, the distribution may not accurately account for external factors that could affect the selection of cookies, such as human error or bias.

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