What would be the best limit to start at?

[jbrennen]

Actually, you probably need a lot more trials than that to have a strong chance of being within 1% of the expected value.

For instance, the number of heads expected in a coin flip is 0.5 with 0.5 standard deviation.

Flip 10,000 coins, and the expected value goes up by a factor of 10,000. The standard deviation goes up by the square root of that, or by a factor of 100.

So flip 10,000 coins, and you expect 5,000 heads with a standard deviation of 50. So +/- 1% from the expected value is one standard deviation. The chance to fall within 1 std. dev. of a normally distributed probability is about 2/3. So if you flip a coin 10,000 times, the chance that it will NOT be within +/- 1% is a relatively high 1/3 probability.

To get +/- 1% to represent 3 standard deviations (giving you about a 99% chance of being within that range), you'd need to flip 90,000 coins.