How Many Hats Must Be Sampled to Guarantee Every Color Is Included?

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Homework Help Overview

The problem involves determining the minimum number of hats that must be sampled from a set of 400 hats, each represented by one of six colors, to ensure that at least one hat of every color is included in the sample. The context suggests a focus on probability and combinatorial reasoning.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the qualitative nature of the term "likely" and suggest quantifying it, potentially using a probability threshold (e.g., P>0.9). There is also a hint about considering the sampling process without replacement and the implications of color representation in the original set.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and questioning how to approach the quantification of "likely." Some guidance has been offered regarding the sampling method and considerations for ensuring all colors are represented.

Contextual Notes

There is an assumption that each color is represented equally among the hats, which may influence the sampling strategy. The discussion also touches on the need for clarity regarding the probability of representation for each color.

neutrino71
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In a set of 400 hats, each hat is represented by one of six colors (brown, pink, red, orange, blue, yellow). You may assume that each color is represented rather equally in the 400 hats.
What is the minimum amount of samples we must take to be likely to have a sample which has a hat of every color?
 
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"likely" is rather qualitative. If can quantify it, say P>0.9, then an answer can be obtained, although the calculation may be a little messy.
 
neutrino: do you have any thoughts about how you might start this problem?
 
neutrino71 said:
In a set of 400 hats, each hat is represented by one of six colors (brown, pink, red, orange, blue, yellow). You may assume that each color is represented rather equally in the 400 hats.
What is the minimum amount of samples we must take to be likely to have a sample which has a hat of every color?
what minimum # of hats must be drawn (without replacement) to have probability 1 of having each hat color represented at least once (if each color is "rather equally represented" in the original set of 400 hats)?
(hint: what minimum # must be drawn to use up all colors except yellow, then choose 1 more.)
 

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