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Given that a mean count of ants on a peony is 25 with standard deviation 47 and mean count of ants on a rose is 58 with standard deviation 97, how do we calculate the probability that the total ant count on 50 flowers is greater than 900?

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    $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ Commented Nov 19, 2015 at 22:48
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    $\begingroup$ What are the 50 flowers (25 & 25, 49 & 1, something else)? Do you want the probability the sum is exactly 900, at least 900, 900 +/- 50, something else? $\endgroup$ Commented Nov 19, 2015 at 22:50
  • $\begingroup$ its not from a text book although its similar in style to a problem that is. information about the 50 flowers is not given. $\endgroup$
    – maryt
    Commented Nov 20, 2015 at 3:48
  • $\begingroup$ Where is it from? What do you mean that "information about the 50 flowers is not given"? You should probably add the tag & follow the procedures outlined in the wiki anyway. Note that if we don't have any information about the distribution of flowers, the question is not answerable. $\endgroup$ Commented Nov 20, 2015 at 16:27
  • $\begingroup$ " Note that if we don't have any information about the distribution of flowers, the question is not answerable". Thank you. you've just answered my question. $\endgroup$
    – maryt
    Commented Nov 20, 2015 at 19:14

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If we don't have any information about the distribution of flowers, the question is not answerable.

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While the question doesn't supply all the needed information (and so gung's answer is correct), we can outline approaches that we could use if we had some more information -- it is likely to be be intended to work in one of only a few ways.

First: I assume that the standard deviations in the question refer to the number of ants not the distribution of the mean count.

Second: I presume we're supposed to assume independence of ant counts across flowers, irrespective of species.

Third: I presume we're meant to assume the total ant count can be approximated by a normal distribution.

Then here's some discussion of what would be involved for the two cases I think are most likely what's being asked about:

  1. If we had fixed counts of flowers -- $n_r$ roses and $50-n_r$ peonies, then the distribution of total ant-count would be based on the convolutions of the individual random variables, but using the basic properties of expectation and variance combined with the normal assumption I mentioned before, we can derive a normal approximation to the total count of ants.

  2. Let's assume we have a random sample of 50 flowers, with some fixed probability, $p_r$ of each one being a rose and $1-p_r$ of being a peony. This is a finite mixture distribution with two components, which for a single flower we can calculate a mean and variance for, and then add 50 of those using a similar approach to that in 1.

[I haven't actually given the calculations for the specific case here because it reads like a self-study question, if somewhat unclearly worded, but at least that's an outline of how to do it under those two plausible sets of assumptions about what's intended.]

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