# probability of an event given two overlapping distributions?

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?

• Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. Commented Nov 19, 2015 at 22:48
• 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? Commented Nov 19, 2015 at 22:50
• its not from a text book although its similar in style to a problem that is. information about the 50 flowers is not given. Commented Nov 20, 2015 at 3:48
• 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. Commented Nov 20, 2015 at 16:27
• " 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. Commented Nov 20, 2015 at 19:14

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

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.]