One of the original questions was: I understand “gambler’s fallacy” where it is mistaken to assume that if something happens more frequently during a period then it will be less frequently in the future. Example: If I flip a coin 9 times and each time I get HEADS, than to assume that it is more “probable” that the 10th flip will be tails is a incorrect assumption.
I also understand that before I begin flipping that coin in the first place, the odds of getting 10 consecutive HEADS is a very big number and not a mere 50/50.
My question is: Is it more likely?, more probable?, more expectant?, or is there a higher chance of a coin turning up TAILS after 9 HEADS?
Physicist: Questions of this ilk come up a lot. Probability and combinatorics, as a field study, are just mistake factories. In large part because single words massively change the difference between two calculations, not just in the result but in how you get there. In this case the problem word is “given”.
Probabilities can change completely when the context, the “conditionals”, change. For example, the probability that someone is eating a sandwich is normally pretty low, but the probability that a person is eating a sandwich given that there’s half a sandwich in front of them is pretty high.
To understand the coin example, it helps to re-phrase in terms of conditional probabilities. The probability of flipping ten heads in a row, , is . Not too likely.
The probability of flipping tails given that the 9 previous flips were heads is a conditional probability: P(T | 9H) = P(T) = 1/2.
In the first situation, we’re trying to figure out the probability that a coin will fall a particular way 10 times. In the second situation, we’re trying to figure out the probability that a coin will fall a particular way only once. Random things like coins and dice are “memoryless”, which means that previous results have no appreciable impact on future results. Mathematically, when A and B are unrelated events, we say P(A|B) = P(A). For example, “the probability that it’s Tuesday given that today is rainy, is equal to the probability that it’s Tuesday” because weather and days of the week are independent. Similarly, each coin flip is independent, so P(T | 9H) = P(T).
The probability of the “given” may be large or small, but that isn’t important for determining what happens next. So, after the 9th coin in a row comes up heads everyone will be waiting with bated breath (9 in a row is unusual after all) for number ten, and will be disappointed exactly half the time (number 10 isn’t affected by the previous 9).
This turns out to not be the case when it comes to human-controlled events. Nobody is “good at playing craps” or “good at roulette”, but from time to time someone can be good at sport. But even in sports, where human beings are controlling things, we find that there still aren’t genuine hot or cold streaks (sans injuries). That’s not to say that a person can’t tally several goalings in a row, but that these are no more or less common than you’d expect if you modeled the rate of scoring as random.
For example, say Tony Hawk has already gotten three home runs by dribbling a puck into the end zone thrice. The probability that he’ll get another point isn’t substantially different from the probability that he’d get that first point. Checkmate.
Notice the ass-covering use of “not substantially different”. When you’re gathering statistics on the weight of rocks or the speed of light you can be inhumanly accurate, but when you’re gathering statistics on people you can be at best humanly accurate. There’s enough noise in sports (even bowling) that the best we can say with certainty is that hot and cold streaks are not statistically significant enough to be easily detectable, which they really need to be if you plan to bet on them.