Thank you for your quick response! Sorry I was a bit slow in replying, but I'm still a bit confused. After reading an overview of binomial distributions, I found a few formulas (and I hope I've used them correctly).

(p^k)(1-p)^(n-k)

n!/[k!(n-k)!]

n = number of events (such as flips of a coin)

k = number of desired outcomes (such as 'heads')

p = probability

The bit I read said the first expression is the probability of each outcome. This makes sense to me. In the case of flipping a fair coin, if we flip 4 times, we get the same result as the reciprocal of 2^4, or 0.0625. Because there are 2^4 possible outcomes, correct?

It said the second expression is the total number of possible desired outcomes. Going back to the example of 4 coin flips, when I am looking for 2 heads, the expression results in 6. In other words, out of 16 possible outcomes from flipping a fair coin four times, 6 of those outcomes have exactly two heads (and therefore two tails), correct?

Finally, it said the product of the first and second expressions is the probability of k out of n ways. This makes sense to me, if I have understood the above. Back to the example of four flips, 6 * 0.0625 = 0.375... this means that if I flip a fair coin four times, I am 37.5% likely to get a head:tail ratio of 1:1, correct?

Please do correct me on any of the above if I am misunderstanding or (unknowingly) misrepresenting what these expressions mean or imply.

Now, the reason I am confused is when I increase the number of flips and still am only looking for a ratio of 1:1, it seems that I get less likely to see that over time. In fact, only with either one or two flips do I get a probability of 50% heads. After that, with each successive flip it seems that the product of the first and second expressions I posted only gets smaller and smaller. This is why I believe I must still not be getting exactly what I'm looking at, if, as you said, the probability will approach 0.5 as the number of flips increase.

Or did I just not increase the flips enough? In other words, does the probability drop near zero, then after a sufficiently large number of flips, begin to go back up towards 0.5? If so, how can I calculate just when that is?

Also, I did not find any calculators that would calculate what I was originally asking about. Either that, or I didn't know them when I was looking at them (since I wasn't sure what some of the variables represented and what some of the solutions meant. And we haven't even touched on the confidence interval yet.

Again, I apologize for being the layperson asking the possibly bizarrely specific question. But I appreciate the help and politeness I have been so far afforded. Thanks again!