Can a coin with an unknown bias be treated as fair?

Question

This morning, I wanted to flip a coin to make a decision but only had an SD card:

Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"?

I thought if I'm just as likely to assign an outcome to one side as to the other, then it must be a fair. But this also seems like a recasting of the original question; instead of asking whether the unknowing of the SD card's construction defines fairness, I'm asking if the unknowing of my own psychology (e.g. which side I'd choose for which outcome) defines fairness. Either way, I think I'm asking: What's the exact relationship between not knowing and "fairness"?

Additional thought: An SD card might be "fair" to me, but not at all fair to, say, a design engineer looking at the SD card's blueprint, who immediately sees that the chip is off-center from the flat plane. So it seems fairness even depends on the subjects to whom fairness matters. In a football game then, does an SD card remain "fair" as long as no design engineer is there to discern the object being tossed?

Answer 1

Here's a pragmatic answer from an engineer. You can always get a fair 50/50 outcome with any "coin" (or SD card, or what have you), without having to know whether it is biased, or how biased it is:

• Flip the coin twice.
• If you get $HH$ or $TT$, discard the trial and repeat.
• If you get $HT$, decide $H$.
• If you get $TH$, decide $T$.

The only conditions are that (i) the coin is not completely biased (i.e., $\Pr(H)\neq 0, \Pr(T)\neq 0$), and (ii) the bias does not change from trial to trial.

The procedure works because whatever the bias is (say $\Pr(H)=p$, $\Pr(T)=1-p$), the probabilties of getting $HT$ and $TH$ are the same: $p(1-p)$. Since the other outcomes are discarded, $HT$ and $TH$ each occur with probability $\frac{1}{2}$.

Answer 2

That is a very good question!

There are (at least) two different ways to define probability: as a measure of frequencies, and as a measure of (subjective) knowlegde of the result.

The frequentist definition would be: the probability of the sd card landing "heads" is the proportion of times it lands "heads", if you toss it many times (details ommited partially because of ignorance: what do we mean by 'many'?)

The "knowledge" approach (usually called bayesian) is harder to define. It asks how likely you (given your information) think an outcome is. As you have no information about the construction of the sd card, you might think both sides are equally likely to appear.

In more concrete terms, say I offer you a bet: I give you one dollar if 'heads', and you give me one if 'tails'. If we both are ignorant about the sd card, then, for us both, the bet sounds neither good nor bad. In a sense, it is a fair bet.

Notice that the bayesian approach defines more probabilities that the frequentist. I can, say, talk about the probability that black holes exist. Well, either they do, or they don't, but that does not mean there are no bets I would consider advantageous on the matter: If you offer me a million dollars versus one dollar, saying that they exist, I might take that bet (and that would 'imply' that I consider the probability that they don't exist to be bigger than 1 millionth).

Now, the question of fairness: if no one knows anything about the sd card, I would call your sd card toss fair. In a very meaningfull way: neither of the teams, given a side, would have reason to prefer the other side. However, obviously, it has practical drawbacks: a team might figure something out latter on, and come to complain about it. (that is: back when they chose a side, their knowledge did not allow them to distinguish the sides. Now, it does)

It the end: there is not one definition of probability that is 100% accepted. Hence, there is no definition of fair that is 100% accepted.

http://en.wikipedia.org/wiki/Probability_interpretations

Answer 3

The SD card is biased (or not), that is, it falls on face A with probability $p$ and on face B with probability $1-p$, for some unknown $p$. If it is "just as likely to assign an outcome to one side of this SD card as to the other", this means that, either one is faithful to the SD card, that is, one answers the result produced by the SD card, or one is unfaithful to the SD card, that is, one answers the result opposite to the result produced by the SD card.

If faithfulness or unfaithfulness are decided independently of the result of the SD card and with equal probabilities, then the final result is fair, irrespectively of the value of $p$.

Note that, if the goal is to answer A or B with equal probabilities, one could as well forget the SD card altogether and simply rely on the perfect source of randomness that was used to choose between faithfulness and unfaithfulness, that is, to encode the result produced by the SD card. Note also that such perfect sources of randomness do not exist in real life (but that very good approximations do).

Is this mathematical factlet the source of your puzzlement?

Answer 4

You could say that there are two unknowns when you flip something: the probability to get side $A$ and the actual result.

So, say you have $3$ coins and $p$ is the probability you get heads:
Coin $A$ is fair ($p=0.5$)
Coin $B$ always gives heads ($p=1$)
Coin $C$ always gives tails ($p=0$)

Obviously, the only fair coin is $A$. But if you don't know which coin you have, say you pick one randomly, then the result is also fair:
$\frac13$ chance to get $A$, then $\frac12$ chance to get heads
$\frac13$ chance to get $B$, you always get heads
$\frac13$ chance to get $C$, you'll never get heads

Therefore, the chance to get heads is $$p=\frac13\times\frac12 + \frac13\times1 + \frac13\times0 = 0.5$$

So, if we assume that the distribution of the chance to get heads of the set of objects you might pick to flip is even, then you can say that the outcome of the SD card is fair as long as you do not, consciously or unconsciously, know or estimate the probability of the card before you select it.

In other words, you rely not on the randomness of the flip but on the randomness of the selection of the object.

Of course, this method won't work for repeated flips and it generates more trust issues compared to a plain coin when more people are involved.

Answer 5

My look at it is this. You are moving probability from one place to the other.

Assume side 1 has probability $p$, and side 2 $1-p$.

If you were to use a fair coin to determine which side of the SD card you get, then you end up with $0.5p + 0.5(1-p)$ of flipping the side you picked(dictated by the coin). This happens to be exactly $0.5$.
EDIT: As this seems to have caused a little bit of confusion, what I mean is if you flip tails on the fair coin, then you'll call tails on the SD card, and the same for heads. This is the probability of flipping the same thing on the coin and SD card.

So the question at the end of the day is, how likely is your choice of side to be as fair as a fair coin toss? Assuming you know NO information, it should be an arbitrary decision, and therefore fair.

The problem is when you try to convince the person you're flipping against that you didn't know which side is better.

In a football game, then, does an SD card become "fair" if none of its viewers can know or imply anything about the construction of the SD card (whether in the stadium or on TV, however improbable)?

In this case actually, I would argue that even if everyone else knew how unfair the coin is, as long as the person calling it has no information about it(you have to assume here that no one hinted it and that he really doesn't know) then it is fair. Again, the same problem as I stated before though. It becomes difficult when you try to convince everyone that he really didn't know.

Answer 6

Imagine taking into account a "fair" coin's initial angular momentum, its initial velocity, initial orientation, the nearby air movements, etc. Assume we can disregard quantum effects and solve for the rest position of the coin in terms of purely classical physics. Assume you'll immediately find and correct any mistakes in your computation.

You should expect to determine perfectly the final state of the coin. It seems that the fairness of a "fair" coin is largely dependent on your state of partial information before the flip. In fact, attempting to ascribe fairness to a coin seems to be a case of mind-projection fallacy.

To be at least a little formal, you can visualize this as the error of defining your fairness function as

fairness(coin)

when instead it should be

fairness(coin, mind)

So, I recommend getting comfortable with the idea of relative fairness. I'm not sure I understand your objection to this idea based on fairness in hindsight. The fairness(coin, mind0) need not equal fairness(coin, mind1), where mind1 is the future state of the mind and thus has the advantage of hindsight.

Answer 7

Yes, it's fair for the first time you use it, then it won't be fair anymore.

If you don't know the possible outcome of something, and randomly map the outcomes of the thing to some result, the first time you use it it'll be fair (since the outcome was unknown by you). After the first time you use it, you've learnt something from it, and that can bias your output.

Another example: take a deck of cards, without looking to the face of any card, shuffle it, and consider that red will be head, black will be tail. You don't need to know if the deck was complete (all 52 cards on it), if it only had black cards, and so on.

Answer 8

The whole idea of a biased coin is basically a convenient fiction useful for making probability examples. It's not really possible to make an object with a coin like aspect ratio significantly biased when tossed. It can be biased when spun (EDIT: or I guess if allowed to bounce). So yes it's actually very close to fair, not just due to your ignorance of it's properties.

Reference: here

Answer 9

I would say that the SD card itself is not a 'fair coin', since a reasonable requirement of a 'fair coin' would be that the probability of landing on each side is essentially 0.5 each.

However, I think the entire process as a whole of assigning an outcome to each side of the card and then flipping the card is a 'fair coin':

The reason why we need coins to make trivial decisions for us is that we can't just cycle through the 2 options in our head, stop at a random point, and choose the option we were thinking of at that point. Not only is this hard to pull off, but you can't be sure that you didn't subconsciously/consciously want to stop on a particular option.

Since the card decides the final result and we don't know the bias of the card beforehand, there is no way that your conscious/subconscious desires would have any way of manifesting themselves when you label each side of the card, and so you label each side essentially randomly, resulting in a 50/50 probability of your two options.

Answer 10

If the conceptual goal of a fair coin is to provide equal chances of two outcomes, then the problem you face is twofold:

1. Is the SD biased?

2. Can you fairly assign the real-world outcomes to the sides of the SD card?

My hunch is that (1) doesn't matter as long as you can achieve (2), but (2) essentially requires a "fair coin", which you don't have.

Answer 11

"If I'm just as likely to assign an outcome to one side of this SD card as to the other, then it must be a fair."

I think you're right with the reasoning. However, the SD card has a definite "front/back", which might be associated with "right/wrong" or "good/bad" in your head. I don't know how strong your bias is, but I would tend to assign the thing I want to the "front" because it's the "good" side. Then if the card is biased, which seems likely, your bias will bleed through to the outcome. I don't think it matters which direction the card is biased, because somebody else who has knowledge of this would recognize the flip as unfair, so it must be unfair on some level.

Answer 12

There are various models of probability that mathematicians may use, such as epistemic, frequentist, or the trivial deterministic model where we assume the future is fixed. From a pure maths point of view picking any of those models is fine, so long as you don't mix models.

For example, I really don't know what the 99th digit of $\pi$ is. So I can say that my epistemic probability that it is 1 is $\frac{1}{10}$. For either a frequentist or deterministic perspective it is clearly 1 or 0, so if we assume that these probabilities are the same we can easily prove that $\frac{1}{10}$ equals either 0 or 1 and then it is easy to show 0=1.

As to which model is "correct", that depends on what you are trying to achieve. If I am trying to make a decision without prejudice I'd be content to pick digits of $\pi$ that I do not yet know. Once I learn the actual digit and my epistemic probability collapses to 0 or 1 it doesn't matter as I have already committed to my decision and moved on. On the other hand, it may be unwise to choose passwords based on digits of $\pi$, even if no-one has yet computed those particular digits. Likewise when Debian chose a random key generator that no-one knew was biased let alone how it was biased it was a disaster for security, despite all possible keys having the same epistemic probability at the time.

Answer 13

As an aside, related to the question but not actually answering it: even if you assign sides fairly and randomly, and keep the assigned sides after the first toss, you still can't do multiple flips and still get the same outcome as a fair coin:

Suppose the probability of a "head" is $p$, and the probability of assigning event $A$ to heads is $1/2$. Then:

$P(HH) = p^2$

$P(TT) = (1-p)^2$

so $P(AA) = P(HH)/2 + P(TT)/2 = (1 - 2p + 2p^2)/2 = 1/4 + (p-1/2)^2$

If the "coin" is fair, then the probability of getting $A$ twice is $1/4$. But if the "coin" is not fair then this probability increases, up to $1/2$ if the "coin" only ever lands one way up.

Answer 14

The card must be fair in both cases, or in no one, because its behavior does not depend on the observer's knowledge:

• Case 1: an engineer knows whether the card is fair, flips it and tells you the result,
• Case 2: you do not know whether the card is fair, and flip it.

Therefore, do not assume that the card is fair in case 2. You need a proof that it is fair.

Answer 15

I'll give it another try:

Assume that the SD card falls on the back with probability $p$ and on the front with probability $1-p$. Further assume that you decide for option $A$ if the SD card falls on its back with probability $q$, and for $A$ with probability $1-q$ if the card falls on its front, and for option $B$ in the other cases. Then $$ P(A) = pq + (1-p)(1-q) = 2pq - p - q + 1.$$

Now if $q=.5$, i.e. you assign option $A$ fairly to one of the card's sides, we get $$ P(A) = p-p-.5+1 = .5, $$

a "fair coin". Does this make sense or answer your question?

Answer 16

TL;DR: carry a true random number generator with known distribution on your person at all times[*].

If there's no correlation between your assignment of outcomes to faces of the SD card, and the SD card's bias, then yes, it doesn't matter whether the SD card is biassed. Saying "no correlation" means it's as if the outcomes were assigned to the faces of the SD card by an unbiassed random choice. So there's your randomness, regardless of what the SD card does. It could be 100% biassed as long as you assign the outcomes to faces randomly. This isn't disturbing :-)

Naively one can argue that if we're capable of assigning outcomes to faces "as if randomly" then we could just choose an outcome "as if randomly". Not so, of course, the point is that any knowledge we have of which outcome will actually happen affects our assignment. For an unbiassed coin flip we cannot have any such knowledge. For a biassed coin flip we're concerned that we might unknowingly act on any information or prejudice we do have. Note that even if we're in some sense trying to get a particular outcome and our information/prejudice is wrong, we still create an unfair choice. It's unfair in the opposite direction to the one we want, but still unfair. That's enough to lose at rock-paper-scissors, and surely the ultimate goal of all probability and game theory is to break even at rock-paper-scissors?

So, you could argue against the existence of that correlation in two ways:

1. Experiment. Observe whether you in particular, or people in general, seem to create any bias of outcomes.
2. Theory. State that since you have no knowledge of which side of the card is favoured, you cannot possibly correlate that to the choice of outcomes.

Experiment 1 seems overkill for this particular case, since the actual outcomes matter -- there might be some outcomes that your brain is capable of assigning in an unbiassed way and others which it is not. A more general such study might be good. Besides, if you could set that up you could probably just fetch a coin.

Theory 2 disappears as soon as you have used the SD card once. At that point you have some information (admittedly not a lot to start with) about its possible bias, so you can no longer claim to be ignorant.

In fact I think argument 2 is dubious anyway. Aside from anything else it's possible that you are subconsciously able to observe from the SD card some feature that (without you realising it) contributes to its bias when flipped. Furthermore that could affect your assignment of outcomes.

If you couldn't see the SD card then you'd be on firmer ground. Suppose that a third party coloured the faces of the SD card "red" and "blue", and you assigned the outcomes to "red" and "blue" before seeing the card. Then my proposed connection between flip bias and outcome bias needs two correlations:

• you need to assign the outcomes to "red" and "blue" based on some intuition of which is more likely
• the third party needs to assign "red" and "blue" to the faces based on some intuition of which face is more likely.

If either of those fails, then the choice of outcome becomes fair.

It starts to sound implausible both those things would hold, but the fact I'm even considering it is because I consider the original scenario plausible, i.e. your original process is not certainly free from bias.

For the football game where the coin is later discovered to be biassed, then provided everybody accepts that there is no way that bias could have affected the call or the person making the toss, I don't think there's a problem. But probably in practice everyone would accept that on somewhat shaky grounds. Good enough for football isn't necessarily good enough for mathematical theory.

[*] von Neumann's construction of a known distribution from an unknown distribution is fine, provided that you can't somehow (intentionally or unintentionally) influence the outcome of the second toss based on the result of the first. You can achieve that in the first round by using a camera, and only examining the result of the first toss after making the second. However if you get a pair and go to a subsequent round then you risk having some information about how to effect that result, which could alter your flipping action. Basically when you don't trust your own subconscious influences you're in deep trouble.

Answer 17

If you toss the SD card, and hide the result.

Then, not knowing anything (nor the result, nor the probability of each side), you pick one side. We can then assume you had a 50% chance of picking a side — as you can not distinguish which side has greater probability.

You then uncover the SD card.

This way, the probability of each side is not taken into account, so if this manner of doing (first tossing, then picking unknowingly) is the same as first picking, then tossing, can we say the toss was fair ?

Answer 18

you have a chance of $p_1$ that you choose a certain side, when you flip it there is a chance $p_2$ that that side comes up

this means that you have a chance of $p_1*p_2+(1-p_1)*(1-p_2) = 2*p_1*p_2-p_1-p_2+1$ to win and $p_1*(1-p_2)+(1-p_1)*p_2=p_1+p_2-2*p_1*p_2$ that you lose.

if neither is a perfect 50% (you are biased to picking a side and the card is biased to land on a certain side) then there will be a bias in the outcome.

Answer 19

It COULD be a fair toss. Prior knowledge of anything but adequate testing is of no help to you. You just cannot establish fairness without testing. It MAY be determined to be fair in the future - AFTER testing. Not knowing doesnt make it fair. Any deviation from determining fairness by testing is pointless. You could only convince someone who didnt understand probablity, and that would only be through trickery.

However, you originally state that you wanted to make a decision, then got distracted by the fairness of the method you chose. That's just procrastination. a fair item for a "coin toss" is not required to make a decision, because the whole point of tossing in that situation is to be aware of your desired outcome before the item shows you the result. In your situation, making a decision and determining the fairness of tossing an SD card are conflicting desires,

Answer 20

If there was no correlation between your (conscious and unconscious) knowledge of the SD card (which includes knowledge of any previous flips) and your choice, then it would be fair. However, this will not be the case.

The unknowing of your own psychology is not enough to eliminate the bias, no matter if the bias is negative or positive.

Answer 21

I feel there are some disturbing implications if the above were true, but I'm unable to convince myself it definitely isn't true.

Why? Is it because you think that the SD card is obviously not balanced, so it cannot possibly be fair? But you said:

Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"?

If you don't know anything about the bias, you don't know. If you think it might not be balanced because of the asymmetrical structure, then you do know something. You don't necessarily know how it's biased, but you know that it might be biased - and fail to yield a 50/50 distribution.

Now we have really moved away from mathematical analysis of an ideal fair coin, and moved on to questioning whether a given real coin (in this case the SD card is taken to be a sort of "coin") is going to behave like the ideal coin.

This problem can be solved by applying hypothesis testing. Your null hypothesis is:

$H_0$: The coin is unbiased, and when flipped a sufficiently large number of times, will come up with one side up 50% of the time, and the other side up 50% of the time.

Then you perform an experiment. Let's say you flip it 52 times, and it comes up with the front up 36 times (69%) and the back up 16 times (31%). Now you must decide, does the null hypothesis still stand? Is it conceivable for such a difference to be generated by chance?

More cogently, you want to ask - what is the probability of this result if the null hypothesis were true? In this case, you can obtain the probability (called a p-value) from the binomial distribution. If the probability satisfies you, then you have no reason to suspect bias in your coin. If it doesn't satisfy you, then it is clearly biased.

You must make a judgement call on what probability is acceptable. Most people use 0.05 (1 in 20 chance).

Answer 22

Why have we over-complicated this? Assuming neither of you knows the bias that the SD card has as to which side it will most likely land on, then you both have an equal chance to pick either "heads or tails," therefore making the toss fair...

Answer 23

As has been pointed out, there are 2 events involved: the decision as to which side of the SD card to associate with which course of action and the flipping event which determines which side lands upwards.

It's normal for those two events to occur in that order. But not necessary: an umpire may toss a coin and conceal the result while the opposing captains call heads or tails. In that event, the outcome of the coin toss is certain at the point at which the heads/tails call is made, but the toss is still considered fair.

Now suppose that the SD card is in fact certain to come down on one particular side (doesn't matter which). If you don't know this in advance, then by analogy with the above, the test is still fair as you don't know that in advance. If it is fair in this case, then it is fair for all values of p

Answer 24

The reality is that few coins are actually truly 50/50. If the center of gravity of the coin does not lie perfectly between the landing surfaces of the two sides then there will be some slight bias toward one side. Differences in the images imprinted on the two sides of a traditional coin normally cause the center of gravity to be closer to one side or the other.

That said, as long as there is no way to discern with the eye which side is more likely to land facing up, then the first flip will always be fair if one person is 'calling' a side for the win. It's the same as if the winner was determined by one person writing either a 1 or a 0 on a piece of paper and folding it up and the other person guessing if the number is a 1 or a 0. There is 100% bias toward the 1 or the 0 after it has been written down but until that bias is known by the person choosing the number the odds are 50/50. The SD card may land with a certain side up 100% of the time but until that bias is known the odds are 50/50.

Answer 25

By definition, an unbiased coin is fair. So, I suspect the answer is "don't know".

Let $F=${"the coin is fair"} and $B=${"the coin is biased"} and $P\left ( B\cup \bar{B} \right )=P\left ( B \right ) + P\left ( \bar{B} \right ) = 1$ (obvious). Then $$P\left ( F | B \cup \bar{B} \right ) = \frac{P\left ( F\cap \left ( B \cup \bar{B} \right ) \right )}{P\left ( B\cup \bar{B} \right )}=P\left ( F\cap B \right ) + P\left ( F \cap \bar{B} \right )$$ But, total probability says: $$P\left ( F \right )=P\left ( F| B \right )\cdot P(B) + P(F|\bar{B})\cdot P(\bar{B}) = P(F\cap B) + P(F\cap \bar{B})$$ So $$P\left ( F | B \cup \bar{B} \right ) = P(F)$$ and $B \cup \bar{B}=${"the coin is either biased or not"} (sort of "don't know"). Or, as other people suggested, the fact that we don't know if the coin is biased or not, has no value for determining if the coin is fair or not.