# Head or Tails: The Question of Determinism and Probability

• B
In summary, the conversation discussed the concept of determinism and how it relates to the results of flipping a fair coin. While the principle of determinism states that under the same conditions, the same causes produce the same effects, the conversation explored how this may not hold true for coin flipping. It was argued that due to random fluctuations and deviations from expected results, the outcome of flipping a coin may not always be 50/50. However, the distinction between "similar" and "same" inputs was highlighted, emphasizing that randomness does not have any bearing on determinism. Ultimately, there was a disagreement over whether the results of coin flipping can be considered truly random or if they are still subject to the principle of determinism.

TL;DR Summary
The principle of determinism states that under the same conditions, the same causes produce the same effects. But when tossing a fair coin, the probabilities of getting heads or tails are equal to 50%. Random fluctuations lead to deviations from the expected results, not probabilities of 50/50.
I have a question that is bothering me. It is commonly accepted that when playing heads or tails with a fair coin and a large number of tosses are made, the probabilities of getting heads or tails are equal to 50% for each toss. However, the principle of determinism, which states that under the same conditions, the same causes produce the same effects, does not seem to apply in this case. In fact, if we toss a coin in a similar manner every time, the principle of determinism, as I understand it, would suggest that we will always get the same side with a probability of 100% rather than 50%.

Some may argue that in practice, it is impossible to reproduce the exact same tossing conditions each time, which can result in random deviations from the expected results. However, random fluctuations lead to random deviations from the expected results, not probabilities of 50/50.

If it's not 50/50, what should it be? 60-40 heads?

TL;DR Summary: The principle of determinism states that under the same conditions, the same causes produce the same effects. But when tossing a fair coin, the probabilities of getting heads or tails are equal to 50%. Random fluctuations lead to deviations from the expected results, not probabilities of 50/50.

In fact, if we toss a coin in a similar manner every time, the principle of determinism, as I understand it, would suggest that we will always get the same side with a probability of 100% rather than 50%.
What does "similar manner" mean? Your statement requires an ironclad definition here to have any meaning.

TL;DR Summary: The principle of determinism states that under the same conditions, the same causes produce the same effects.

In fact, if we toss a coin in a similar manner every time, the principle of determinism, as I understand it, would suggest that we will always get the same side with a probability of 100% rather than 50%.
Notice that you made a small but critical change. In the first paragraph you used the word “same” and in the second paragraph you used the word “similar”. The principle of determinism does not make any claims about “similar”. A deterministic system will give the same output for the same input, but similar input may produce dramatically different output.

DennisN, russ_watters, PeroK and 3 others
PeroK said:
If it's not 50/50, what should it be? 60-40 heads?
I expected a result closer to 100-0 than 50-50 since we are talking about a deterministic event.
and if we take into account the fluctuations of the initial conditions, and if we consider that the effects are random, then I expect random results but not 50-50

Dale said:
Notice that you made a small but critical change. In the first paragraph you used the word “same” and in the second paragraph you used the word “similar”. The principle of determinism does not make any claims about “similar”. A deterministic system will give the same output for the same input, but similar input may produce dramatically different output.
I explained the similar word in the body of the question:
"Some may argue that in practice, it is impossible to reproduce the exact same tossing conditions each time, which can result in random deviations from the expected results. However, random fluctuations lead to random deviations from the expected results, not probabilities of 50/50."
So you agree that heads or tails games in reality cannot give results close to 50/50 but rather random results?

Motore
In a way you are right.
Suppose that you get H heads and T tails. Then as you toss the coin the difference between H and T of |H-T| usually grows.

But |H-T| / (H+T) = P shrinks. As you toss the coin then the probability that P is, say, less than .00000001 grows. Toss the coin enough times and this probability can be as high as you like.

So you agree that heads or tails games in reality cannot give results close to 50/50 but rather random results?
In reality flipping a fair coin will give you heads 50% of the time and tails 50% of the time. That's because the initial conditions are uniformly distributed and thus also the outcomes will be uniformly distributed.
Well, not quite..:

I explained the similar word in the body of the question:
Yes, but that explanation doesn’t change the objection.

The “similar” vs “same” distinction means that your arguments do not and can not have any bearing on determinism. Because you are dealing with only similar inputs rather than same inputs randomness or non-randomness gives no insight on determinism or non-determinism.

So my issue with your OP is about the attempt to tie randomness to determinism and invoking the principle of determinism.

So you agree that heads or tails games in reality cannot give results close to 50/50 but rather random results?
I disagree. You have expressed this as a dichotomy. The “close to 50/50” or “random results” are not mutually exclusive. In a fair coin both are true. A fair coin is both random and close to 50/50.

hutchphd
It is probably possible to program a robot to flip a coin with enough repeatability to deviate from 50/50 or even potentially give the same result every time, especially if you make the flip small. But it does not take much deviation from exactly the same for randomness to become dominant.

None of that says anything profound about determinism or randomness.

DennisN, Vanadium 50, Bystander and 1 other person
I think there are people who can flip the bigger coins with quite a bias.

russ_watters
So you agree that heads or tails games in reality cannot give results close to 50/50 but rather random results?
The is the law of large numbers which indicate a convergence to mean as N gets large. The mean here is $$\frac {heads-tails} {heads+tails}=0$$
There are some nuances which I will get wrong......

hutchphd said:
The is the law of large numbers which indicate a convergence to mean as N gets large. The mean here is $$\frac {heads-tails} {heads+tails}=0$$
There are some nuances which I will get wrong......
$$lim_{\text{tosses} \to \infty} \frac{\text{heads}\ - \text{tails}} {\text{heads}\ + \text{tails}} = 0$$ is an attractive hypothesis. However, it is untestable.

I am unsure what that means in this case.

It just means that you cannot do an infinite number of tosses and there is no finite number of tosses that guarantees convergence to within some specified deviation from 0.

jbriggs444
russ_watters said:
It is probably possible to program a robot to flip a coin with enough repeatability to deviate from 50/50
Or a human, if you used the right coin:

It might take a few humans to life it. "Flipping" might not be the right word.

But I am trying to figure out the physics content of this thread. It seems to be that an approxiimation is involved. Why is this better than a frictionless plane or massless pully.

Bystander, Dale, hutchphd and 1 other person