[p-orbs]

I know the definitions of the two terms, but I need some guidance.

Does the sample amount affect precision?

If I had a big bag of coins containing different amounts of pennies, dimes, nickels and quarters. At first I grab one handfull of coins, count them, and then repeat twice.
Second, I grab two handfulls of coins, count them and then repeat twice.

I would think the second batch would be closer to the true composition in the bag, and therefore give more accurate results? At least in theory.
But when it comes to precision, which method is more precise? Are they just as precise, with no regard to the sample size? I argue with myself back and forth, can't seem to wrap my head around this 

Thanks!

[mjc123]

The accuracy would be the same, but the second method would be more precise.

Well the term "handful" is pretty imprecise to start with. Let's say a handful is 20 coins. Suppose the fraction of pennies in the bag is P. The distribution of the number of pennies in your handful is F(n) = Pⁿ(1-P)^20-n*20!/(n!(20-n)!). This is a binomial distribution.
The expectation value of n (what you would expect to be the mean value of many trials) is 20P, and the variance (square of the standard deviation) is 20P(1-P). The expectation value of p, the fraction of pennies in your handful, is P, and the variance is P(1-P)/20.
Now suppose you take two handfuls, i.e. 40 coins. E(n) is 40P, V(n) is 40P(1-P), E(p) is P , and V(p) is P(1-P)/40. The expectation of p is the same - p is an unbiased estimator of P, which doesn't depend on the sample size. The variance of p decreases with increasing sample size, so p is more precise for larger samples.
In either case, if you did the trial 3 times and took the mean value of p, the expectation would be the same (P), but the variance would be divided by 3.

If you had a source of systematic error, e.g. the heavier coins sank to the bottom of the bag, this would probably be reflected in an effective probability of drawing a penny, P', different from the fraction of pennies in the bag, P. Both your methods would estimate P', so neither would be more accurate.

[p-orbs]

Whoa this was complicated! 
But how come the accuracy is the same?
If the distribution of pennies in the bag is i.e. 50%, and I take a small sample (one handful) from the bag, do this three times, and come up with a mean value. Won't a larger sample size (two handfuls) be more representative (accurate) of the content in the bag?

[mjc123]

I think you need to revise the (scientific) definitions of accuracy and precision. You seem to be using "accuracy" in a looser sense, which confuses accuracy and precision.
I have just demonstrated mathematically that the accuracy is the same. The expectation of the variation of p from the true value P is zero, whatever the sample size. The precision is statistically defined as the reciprocal of the variance, so the second method gives a more precise estimate of p.
Consider the graph below. The curves give (normal) probability distributions of a variable x which is an estimate of a quantity whose true value is 5. The blue curve represents an estimator that is accurate but less precise; the green curve one that is more precise but less accurate. The red curve is both accurate and precise. The blue and red curves are analogous to your first and second methods; the second is more precise but not more accurate, since both have an expectation value of 5. There is a higher probability of a value of x from the blue distribution being further from this value than a value from the red distribution, but this is the definition of precision, not accuracy.