So I've had an interesting breakthrough on this question. Instead of attacking the question head-on, I think I have found the solution indirectly by using the formula for calculating sample sizes.

Scientific studies often rely on surveys distributed among a sample of some total population. Your sample will need to include a certain number of people, however, if you want it to accurately reflect the conditions of the overall...

www.wikihow.com

When using this formula, you typically arrive at an answer where you would say something like "With a 95% confidence level, the sample has a 47% chance +/- 5% of having the same ratio as the total population".

So here is what I did with the question above. First, I assume a very large total population, which means, according to that page, you can use the simplified version of the formula:

(z^2 * p(1-p))/e^2

z is the z-score, (not calculated the way used in a previous post!). Instead you determine a confidence level, and from that a z-score.

p is the standard deviation, which is typically 0.5 unless you have reason to assume a different value

e is the margin of error, which you determine.

Running this through the formula gives you a requirement on the minimum sample size.

So I was experimenting with this. My ratio in the sample size for green is 56.26%. I said, let's make the confidence level 95% since that's typical, and that means a z-score of 1.96. I said, let's make the margin of error +/- 5% because that's fairly typical. When we plug our numbers in, we get:

sample size = (1.96^2 * 0.25)/(0.05^2) = 384.16. My actual sample size is larger, so I know that any statement of the accuracy of the sample with this calculation will be less than reality. So what I have in this case is a statement: "With greater than 95% confidence, the ratio in the total population is between 51.26% and 61.26%"

But the lower end of that is greater than 50%. Bingo! I can now say "With greater than 95% confidence, the ratio in the total population is majority green" or to put it another way "There is greater than 95% probability that the ratio in the total population is majority green"

So the procedure for doing this would be:

1. Of the sample with 2 types represented, identify the type (X) with the majority.

2. Choose a margin of error such that the ratio of X in the sample minus the margin of error is still greater than 50%

3. Choose an initial confidence level (I choose 95% to start with) and get the associated z-score.

4. Run the numbers through the formula.

5. If the indicated required sample size is smaller than our actual sample size, then we're done. Our confidence level is less than the probability that X is majority in the total population.

6. If the indicated required sample size is larger than our actual sample size, lower the confidence level (I'd say go in 5% increments) and go to step 4.

In retrospect, I could have picked a margin of error of 0.06 and still met the requirements. This would have lowered the required sample size, and I wonder if I could have compensated by making the confidence level 99%.

I think this works, although it doesn't get you an exact probability (instead you get an "at least"), but it's still meaningful. Being able to say something is somewhat greater than 95% is about as good as saying it's 95%, in most cases.