How exactly does probability change, if you have a history of events in a binomial distribution?

Question

As a quick introduction, remember the Monty Hall problem: You have a choice of three doors: One is good, two are bad. After picking one door, another is opened revealing to be a bad one. With that in mind, the probability of picking the good door now becomes 66% by switching. However someone who has no idea of the first pick will see two doors, assuming both to be 50%.

That being said, if you have dropped a coin to be tail nine times, is the probability of the tenth drop still 50/50, or is there a high probability that it will be head, so when using the cumulative binomial distribution, you would get another result than the 50% head/tail options.

The question is: When exactly does probability get affected by historical events? (Where by 'historical events' I mean the statistics of the events that happened)

Answer 1

The Monty Hall problem isn't about a choice between $2$ doors; it is about the choice between a door and the better of $2$ doors.

In context, the past event of the game show host revealing what lies behind a door is really just a deceptive way of telling you which of the $2$ unpicked doors is better. Hence, a future event of you switching doors is just a switch from selecting $1$ door to selecting the better of $2$ doors.

The reason that the past event affects the future event is because of a change in choices: in the past, we were choosing between $3$ equivalent doors; in the future, we are choosing between a door and the better of $2$ doors.

When we flip a coin and flip it again, and again, our choices do not change: we are still choosing between a head and a tail.

That is the distinction.

Therefore,

The probability of an event occurring will change because of historical events if and only if the historical events change our choices.

Answer 2

The Monty Hall strategy works because the car does not move after you make your first choice. There is no "next flip of the coin."

When you make the decision to switch, you are making a bet on an event that had already happened (the car was placed behind a certain door) before you even made your initial choice of doors.

The "historical events" in this case are the events concerning where the car was placed at the start of the game and which door Monty chose to open. There are constraints between those two events (Monty cannot open the door the car is behind) that make them not independent of each other. Hence you may be able to (and in fact can) use information about one event to obtain information about the other event.

In the case of two coin flips, the flips are independent. That is why information about one gives you no information about the other.