I'm trying to answer questions in the theme of the following:

1) What's the probability that over 100 flips, a fair coin will ever have a score of +8 heads?

2) What's the probability that over 100 flips, a coin with a bias of 53% heads will ever be -4 heads?

3) What's the probability that a coin with a bias of 53% heads will be tied heads-to-tails after 100 flips?

How do I solve these sorts of problems?

Binomial distribution.

There are on-line calculators that can be used or Excel. When you say +8 you have to specify exactly 8 or >=8. Same for exactly -4 or <=-4.

The "ever" aspect of the first two questions adds a wrinkle but it is not too difficult to overcome.

Q3 is the aforementioned Binomial PMF. There are C(100,50) desirable permutations, each with probability (.53*.47)^50, for an overall probability of .066457

Q1 and Q2 are finite-time gambler's ruin problems and they are indeed wrinkly compared to Q3.

A1 = sum_k[8 to 54] of (8/k)*C(2k-9, k-8)*.5^(2k-8) = .42616

A2 = sum_k[4 to 52] of (4/k)*C(2k-5, k-4)*(.47^k)*.53^(k-4) = .51792

With infinite flips, A1 would be 100% and A2 would be (47/53)^4

Good point. I ignored the "ever" and heehaw provided the method it include it

Thanks for the responses.

With infinite flips, are you saying there will never be a 100 flip string where it's -4 heads?

With infinite flips, (47/53)^4 or 61.84% is the chance that the overall score will ever be -4 heads.

If you were to reset the score after every 100 flips, then you'd be 100% to reach a -4, ie it's 100% that there will be a series of 100 flips containing a -4 subtotal. Each disjoint series of 100 flips is independent and therefore has the same .51792 probability of -4, so of course with infinite repeated attempts at a 52% chance it's sure to hit.

When not resetting the score, Heads' lead grows with time and -4 becomes harder to reach. The probability of reaching -4 amounts to a sum of ever-decreasing probabilities, making it different from monkeys typing Shakespeare.

Can this be why some number theory conjectures will never be proven? No one seems to know.

God knows. But God is a moronic concept invented by diseased people for their own nefarious ends. God is ignorant of mathematics. Numbers are an illusion, like getting to heaven after you die.

my sense is that OP's use of "ever" really meant "how often does it happen?"

that's how i read it in relation to the rest of the post.

and definitely, OP should check out "binomial calculators"

If one is familiar with normal distribution probabilities (I.e. probability of at least 1SD greater than average is about 2/3, 2SD greater is about 95%, 3SD about 99%), you can get a decent estimate of the binomial probability by using a normal distribution with an average of N/2 and a standard deviation of sqrt(N)/2 if N is large enough. It may not be a gray approximation for N=100, but it gets better for larger N. For biased coins the same approximation can be used except that the average is pN and the standard deviation is sqrt(Np(1-p)) where p is the bias probability (0.53 in the OP).