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I have a population (n=1000), and I know that there are several groups (sub-populations) within it (k=7). This is a multinomial experiment. How do I calculate the minimal N required for drawing at least 1 sample from each group? It is similar to having an urn with 1000 balls inside, each ball has one of 7 colors (blue, green, etc.). What is the minimal number of balls I need to extract from the urn (without replacement) in order to see all 7 colors?

user9446044
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    can you pls define what you mean by a multinomial experiment? Few additional details about your objective and design would not hurt either – DrGo Sep 12 '21 at 05:17
  • The goal of my design is simply to look at all the available groups in the urn, and be 95% certain that there wasn't another group (color) that I missed. – user9446044 Sep 12 '21 at 06:49

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Assuming equal distribution of colors (ie ~143 balls per color), I believe you need to draw about 19 balls. My logic is that there is 0.143 chance of drawing a ball of certain color (the event). The probability that an event has not happened in n trials is (1-0.143) raised to n. And since you desire 95% probability of the event (or 0.05 not happening), (1-0.143)raised to n = 0.05. Solving for n, n= log(0.05)/log(1-0.143)=19.4

DrGo
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  • That applies for only one of the groups in the population; however, I need to know the probability to draw *at least one ball* from each color group. – user9446044 Sep 14 '21 at 05:36
  • P of drawing at least one ball of a certain color = 1-(P of drawing no balls of that color). I am making the assumption that not seeing one color has nothing to do with not seeing any other color. This is not 100% accurate since the P of seeing a certain color increases slightly when we draw a ball of a different color (if we sample without replacement), but cannot think of an easy way to incorporate that. My apologies. – DrGo Sep 14 '21 at 17:53