# Conditional Probability without replacement and with unique picks

Say I have a bag of 1000 balls. There are 50 uniquely coloured balls, each with known frequency. Every time I pick a ball out, whatever colour it is, all of those coloured balls are then removed from the bag. What is the probabilty of picking a certain colour ball in 20 picks.

Applicability: I work in finance. There is an investment package that allows the customer to choose 20 stocks. We have the data that shows how many customers have this package and the amount of times a certain stock is chosen. For example, say I have 1,000 customers who use this investment package, 550 of them choose NTFLX as one of their 20 stocks.

Goal: I am looking for a way to determine the probability of a stock being picked as one of the 20. All of the probabilities are known for each possible stock choice, I suppose the ultimate goal would be to figure out the probability of being chosen algorithmically.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented Mar 5 at 19:42
• Asking for probability from scratch or after red has been removed ? And what is the point of this problem ? Does it have any practical parallel you have in mind ? Commented Mar 5 at 19:44
• Are you assuming your customers choose their $20$ stocks independently at random? If so, we can calculate the probability you want. But it seems a rather strange assumption to make. If that's not what you mean, then you know the probability for NTFLX since you know how many customers chose it. Please edit the question to clarify. Commented Mar 5 at 20:11
• I suggest: simulate the process. As others have remarked, it is awfully hard to believe that your clients choose stocks uniformly at random. A simulator will let you play with the probabilities. You can try various models to see if any of them capture reality.
– lulu
Commented Mar 5 at 20:38

## 1 Answer

Suppose you have $$n$$ balls, $$m$$ colors and you want $$k$$ picks. In your case, $$n=1000$$, $$m=50$$ and $$k=20$$.

For each color $$i$$, let $$x_i$$ be its frequency. It is assumed then that the sum of the $$x_i$$'s is equal to $$n$$.

The probability of getting the colors $$(c_1, c_2, \ldots, c_k)$$ in a pick in that order is equal to $$\frac{x_{c_1}}{n} \cdot \frac{x_{c_2}}{n-x_{c_1}} \cdots \frac{x_{c_k}}{n-(x_{c_1}+\cdots+x_{c_k})}$$

The desired probability is then the sum of the probabilities (as above) over all the tuples that contain the fixed color.

Without assuming anything else, that's the further I can get, as the probability of two tuples with the same elements is not necessarily the same. Hope that helps!