Probability Problem - Need help

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nkat225

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Mar 25, 2021
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Hello Everyone,

I need help solving the following problem:

A bag contains 2,000 marbles of 100 Colors, 20 marbles/color.
After drawing out 800 marbles from the bag.

What is the expected number of colors (% of the 100 colors) assuming an equal probability of drawing a marble?
 
nkat225 said:
Hello Everyone,

I need help solving the following problem:

A bag contains 2,000 marbles of 100 Colors, 20 marbles/color.
After drawing out 800 marbles from the bag.

What is the expected number of colors (% of the 100 colors) assuming an equal probability of drawing a marble?
Click to expand...
Choose - with replacement or without replacement?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
I think the problem can be expressed as follows:

There are 2,000 balls of 100 colours; 20 balls of each colour.
These are split, by some random process, into two portions, one of 1,200 balls and another of 800.
What is the most probable number of colours (100?, 99? 98? ...) with at least one ball in the 800 ball portion?

A difficult problem because the combinations are complex and the factorials lead to large numbers.

As a start, consider the 20 balls of a particular colour eg red. What is the probability that none are in the 800 ball portion? Because the number is small compared to the total number of balls, we can can approximate by taking the probability of each red ball being in the larger portion as 1200/2000 = .6. Probability that all red balls are in the larger portion is .6^20, = approx 3 in 100,000, very small. So it is almost certain that at least one red ball is in the 800 portion.
Even with this approximation it is difficult to calculate for the other 99 colours (the probabilities are not independent) but it would seem very likely that every colour is represented in the smaller portion.
 
Subhotosh Khan said:
Choose - with replacement or without replacement?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
Click to expand...

My apologies, subhotosh khan. I did read the rules of posting, just not thoroughly enough apparently. I should have mentioned that this is not a homework problem so that's probably why the instructions look incomplete. My team and I were trying to simplify the problem we have so we missed details I guess. To answer your question, it's without replacement.

Thank you JayJay for your input. I think you have reworded the problem correctly. Another way to state the problem is, given the initial conditions, what is the most probable amount of balls I need to pick in order to have at least 1 ball of each color. We are trying to find a formula to connect the percentage of total quantity picked to the % of colors picked within an error margin. We have been confusing ourselves with the large numbers we were coming up with. I think at this point we are interested to know what formula to use to attempt reaching an answer. We are all engineers who haven't touched on a complex probability problem in a long time.

Again, I apologize if this forum is strictly for students, we were just trying to reach out to someone with a Math background to help orient us in the right direction. If this is not the place, it would be much appreciated if you could redirect us to other forums if you know any.
 
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nkat225 said:
My apologies, subhotosh khan. I did read the rules of posting, just not thoroughly enough apparently. I should have mentioned that this is not a homework problem so that's probably why the instructions look incomplete. My team and I were trying to simplify the problem we have so we missed details I guess. To answer your question, it's without replacement.

Thank you JayJay for your input. I think you have reworded the problem correctly. Another way to state the problem is, given the initial conditions, what is the most probable amount of balls I need to pick in order to have at least 1 ball of each color. We are trying to find a formula to connect the percentage of total quantity picked to the % of colors picked within an error margin. We have been confusing ourselves with the large numbers we were coming up with. I think at this point we are interested to know what formula to use to attempt reaching an answer. We are all engineers who haven't touched on a complex probability problem in a long time.

Again, I apologize if this forum is strictly for students, we were just trying to reach out to someone with a Math background to help orient us in the right direction. If this is not the place, it would be much appreciated if you could redirect us to other forums if you know any.
Click to expand...
That's a horse of a different color! You say:

given the initial conditions, what is the most probable amount of balls I need to pick in order to have at least 1 ball of each color:​

If you pick 1981 balls (99*20 +1), you will be assured that you have picked at least one ball of of each of the 200 colors.
 
Last edited by a moderator:
Subhotosh Khan said:
That's a horse of a different color! You say:

given the initial conditions, what is the most probable amount of balls I need to pick in order to have at least 1 ball of each color:​

If you pick 1901 balls, you will be assured that you have picked at least one ball of of each of the 200 colors.
Click to expand...

I guess using "at least" was not appropriate here.

I think JayJay still has the best explanation of the problem. I was just trying to pinpoint that I am more interested in the relation of % quantity picked with % of color.

Let me use the exact problem terminology rather than the previously mentioned in the hope to be less confusing.

I bought 2,000 tops of 100 styles, with an average quantity of 20 tops from each style.
I will be receiving the tops in random batches. Knowing I will be receiving 40% of the tops in the first batch, is there a way to predict the % of styles I will be receiving? Is a prediction possible in this case or additional parameters/assumptions need to be taken into consideration?
 
We help others, but give priority to students.

I’ll think about your problem.
 
The probability that at least one color is missing is

[MATH]\dbinom{100}{1} * \dbinom{1980}{800} * \dbinom{20}{0} \div \dbinom{2000}{800} = 100 * \dfrac{1980!}{800! * 1180!} * \dfrac{800! * 1200!}{2000!} < 1\%. [/MATH]
The probability that some specific number of colors is missing will be much smaller than that.
 
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