[Math] I can’t understand how probability makes sense

probability

I have a lot of questions regarding probability. Please forgive me if I have made mistakes.

  1. I actually tossed a coin 200 times. 54% of the time it landed on heads and 46% it landed on tails.

    What is the reason that there is a fair chance of the coin landing on heads or tails?

    Is it the randomness that causes this?

    If so, then in a purely random experiment would the result be 50-50?

    Even though probability only projects the likelihood of an event, why are the outcomes in favor of this projection?

  2. If I eliminate all the external factors during a coin toss, like air resistance, the coin is tossed in a vacuum chamber, the force to flip the coin is fixed,etc will the experiment still be random? Or will I be able to predict the outcomes?

If the outcomes are indeed predictable, will the experiment be still random if I add a single atom into the chamber? If not at what point does it become random again?

Best Answer

j4nd3r53n's answer is great, and the starting point of mine.

What is the reason that there is a fair chance of the coin landing on heads or tails?

Probability can be thought of as a measure of uncertainty. When a person flips a coin in the real world, there are many, many factors that go into whether the coin flip will result in heads or tails: the force with which the flipper flips, the exact spot on the coin the force is centered, the exact vector of the force, the height difference between where it's flipped and stopped, deformations and imperfections in the coin, the wind, etc.... We say it's 50/50 because so many of those factors are wholly outside of the flipper's control and many that are theoretically within their control (eg., the force they use) are incredibly sensitive to tiny differences. Since humans out in the world can't choose which result they want, we have no information about which side will face up after the flip: we're completely uncertain, so neither choice is better than the other.

If I eliminate all the external factors during a coin toss, ... will the experiment still be random?

On the extreme other end, it is entirely possible to consider a machine that is designed to flip a coin just so so that it always lands heads-up, provided the coin can be placed into the mechanism correctly (eg., it might need to be heads-up to start, with the face angled just so). That machine could well flip the coin thousands of times and get heads each time. The difference here is that all of the factors that affect how the coin flies through the air are controlled, removing the uncertainty of the coin's trajectory. Or, more accurately, the variation now lives within the machine: whether the flipping arm malfunctioned or the case cracked or the landing pad wore down enough that the bounce is now "wrong". In removing sources of uncertainty, we've increased the probability of getting heads.

Is it the randomness that causes [a fair chance of the coin landing heads or tails]?

It's the other way 'round: the fair chance of the coin landing heads or tails is what results in the coin toss being random.

... in a purely random experiment would the result be 50-50?

In the real world, probably not: there's some evidence that heads will result about 51% of the time (partly due to the heads and tails sides not being evenly weighted, though there are other factors).

Here are the broad strokes of their research:

  1. If the coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. (If it starts out as heads, there's a 51% chance it will end as heads).
  2. If the coin is spun, rather than tossed, it can have a much-larger-than-50% chance of ending with the heavier side down. Spun coins can exhibit "huge bias" (some spun coins will fall tails-up 80% of the time).
  3. If the coin is tossed and allowed to clatter to the floor, this probably adds randomness.
  4. If the coin is tossed and allowed to clatter to the floor where it spins, as will sometimes happen, the above spinning bias probably comes into play.
  5. A coin will land on its edge around 1 in 6000 throws, creating a flipistic singularity.
  6. The same initial coin-flipping conditions produce the same coin flip result. That is, there's a certain amount of determinism to the coin flip.
  7. A more robust coin toss (more revolutions) decreases the bias.

-- source

If the outcomes are indeed predictable, will the experiment be still random if I add a single atom into the chamber? If not at what point does it become random again?

Basically, the result will be as random as the unknowns or un-predictable-s allow/force it to be. With the coin flipper machine, a single atom is almost certainly not going to affect the path of the coin, but thermal expansion of the coin or the flipping arm might; a well-timed power surge or failure to the flipping arm is certain to add uncertainty to the flip.

I actually tossed a coin 200 times. 54% of the time it landed on heads and 46% it landed on tails.

For some more theoretical/math-ey bits, Wikipedia's Law of Large Numbers page has some good information and pointers. Extremely basically, it says that "the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer to the expected value as more trials are performed." Which is to say, were you to keep flipping the coin, you'd get closer and closer to that 51/49 ratio. 200 trials is in the realm of a large number of trials, but it's on the low end.

This is not to be confused with the semi-serious Law of Truly Large Numbers, which states that "with a large enough number of samples, any outrageous (i.e. unlikely in any single sample) thing is likely to be observed" - flip the coin 1,000,000,000 times and getting 100 heads in a row becomes pretty likely.

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