What’s the probability of 101 heads over non-consecutive tosses? Imagine there’s this guy flipping a coin every day at every hour and I occasionally pass by him and record the tosses. The first 100 times the tosses are ALL heads. OK that’s weird but it’s guaranteed the coin is perfectly balanced.
Now what’s the probability the 101st time I pass by I will observe an head again? Is it really just 0.5?
 A: The probability of the next toss being heads is $1/2$. Coin tosses are what we call independent, meaning that if I toss the coin at different times, then the outcome of one trial won't affect the other. You have, however, hit on one of the oddities of conditional probability that a very unlikely event like "my friend got 101 heads in a row" seems on its face very unlikely (a $2^{-101} \approx 3 \cdot 10^{-31}$ chance) is relatively likely given what you already know (that the last 100 tosses were heads). It's irrelevant whether your observations were "consecutive" or not.
For an idea of why your observations on the past probably shouldn't matter, consider the following thought experiment: You're at the casino and you just sat down at a blackjack table. There's one other guy at the table. Do you bother to ask him how he's been doing? Well, if you're a perfect sociopath who doesn't care about etiquette, then no, you probably don't. Maybe he's been on a hot streak. Maybe he's been having awful luck. But that makes no difference to you. After all, whatever kind of night he's been having, the dealer just put the cards back in the shuffler and dealt a new hand. Whatever may've happened before is irrelevant. That is, of course, assuming that you trust the shuffler to be fair, but you said in your question that "it’s guaranteed the coin is perfectly balanced," so that seems apropos.
