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I think this is rather an English language question, but I've asked this in ell.sx, and a person there insists that this is a concern of mathematics.

Let's say I have $k_g$ green, and $k_r$ red balls. I want to select one, randomly, but with a bias towards the red balls, as follows;

  • Each of the $k_g$ green balls have $p_g$ probability of getting chosen.
  • Each of the $k_r$ red balls have $p_r$ probability of getting chosen.

Obviously (I hope), $k_g p_g + k_r p_r = 1$.

Now, without giving any of the lengthy explanation above, and without saying $P_g = k_g p_g$, I want to summarize all this information with the following paragraph:

The bag I have consists of $k_g$ green, and $k_r$ red balls. The setup is so arranged that the probability of choosing a green ball is $P_g$. Same color balls have the same probability of getting chosen.

Is the part written in bold in the above paragraph any ambiguous? If it is, how else should I say it?

To clarify: What I'm trying to say is (1) "the chances of a selected ball being green is $P_g$", and not (2) "each green ball has a $P_g$ chance of getting chosen". I find the sentence (1) much more clear, but it is rather a weird way to put it.

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I would consider your phrasing ambiguous; both interpretations (1) and (2) are plausible readings without more context. I think your version (1) (or a minor variation of it) is a perfectly good phrasing to use, actually. I would move to the forefront the condition that different balls can have different probability, though, since typically when describing this sort of situation, it is assumed that all balls are equally. So, I might write your paragraph as something like the following:

The bag I have consists of $k_g$ green, and $k_r$ red balls. When I draw a ball from the bag, balls of the same color have the same probability of being chosen, but balls of different colors may have different probabilities. The probability that the selected ball is green is $P_g$.

Another phrasing you could use is something like "the total probability of any green ball being chosen", where the word total clarifies that you mean the cumulative probability for all the green balls together, instead of one at a time.

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