What’s the mathematical law around this?
In: Mathematics
Let me try out this explanation and see if it makes sense.
Imagine a world where you and your family, for generations, have done The Ritual where, when you turn 18, you flip The Coin. The coin has been passed down, parent to child, since the days of the Roman Republic. It’s your 18th birthday, and YOU get to be the exact 100th person!
And right now there have been 50 tails and 49 heads. Will your coin flip be influenced at all by coin flip from someone you’ve never met, and who died in the plague in 1666?
Answer: no, that’s ridiculous.
But from a logic standpoint, it’s the same problem. There’s nothing magic about the “100 flips” that you’re doing; to you they might be an experiment, but to the coin it’s just another random flip. It’s only us humans and our constant need to turn random events into a narrative (looking at you, Game of Thrones!) that makes us think that the 100th coin would be constrained.
In probability, the general rule is that as the population you are looking at increases, the distribution inside that population will more reliably resemble the statistical distribution.
For instance, if 10% of people are left-handed and you are looking at a population of a billion people, you can very safely say that there will be roughly 100 million left-handed people, with only a very tiny variation from that, probably hundredths of a percent.
If you are dealing with a population of 10 people, on the other hand, it is not that unlikely that you will have no left-handed people or two left-handed people, and those situations represent significant deviations from the statistical distribution.
The sub-discipline of probability that has to deal with these differences from statistical ideals that we see in small populations is called stochastics. Any chance you can find math that gives you some insights into just how big a deviation you would expect in any given population size, for instance.
Because every flip starts the probability over. Say you flip the coin 5 times and it become heads 2 times and tails 3 times. You flip it a sixth time if every flip was connected to the last probability wise it would end up being heads. But since each action of the flip starts back to 50/50 heads and tails both have the same probability to occur each time.
Flipping a coin you have two sides. Therefore you have a 50% chance of getting either side on each flip.
Each time you flip the coin you still have two options. Each coin flip is independent of previous and future coin flips. Therefore the probability of a single event has no impact on the results of multiple events.
As a comparison say you have 5 red marbles and 5 blue marbles in a bag. Reach in the bag and pull out a marble you have a 50% chance of getting a red or blue marble.
Say its a red marble.
Now reach in and pull out a marble. Since 1 red marble has been removed you now have a 4/9 chance of getting a red marble and a 5/9 chance of getting a blue marble. These are dependent events because the results of one event impacts the probability of future events.
Say you had done 99 flips and somehow got 49 heads and 50 tails. There’s still only 50% chance of getting the final head to make it 50:50 and an equal chance of going to 49:51. That’s because each flip isn’t affected by what has gone before. **Before you start,** 50:50 is the most likely outcome (just) but as you build up results whatever you’ve got so far is history and doesn’t guide future ones towards evening out the numbers.
As well as probability and math, I believe the tail side has a slight sliver more weight on it than the head side. This skews some of the flips, possibly by 1 or by 0.5 in a large sample size, and why tails is sometimes more “lucky” than heads.
Because there is always some variance. You’re quite unlikely to get *exactly* 50 heads and 50 tails (~7%), but you’re very likely to get *around* 50 heads and 50 tails. The more you throw your coin the “error” grows slower than the overall ratio, so even if you’re not exactly on the nose with 50:50 the overall ratio between heads and tails will converge closer and closer to 50:50
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