Predicting the outcome of the next coin toss?

Question

Tossing a fair coin is i.i.d. Let's assume that I have that:

Coin = {H,H,H,T,T}

How would you guess the next coin and why?

PS: I looked for Bernoulli Distribution but couldn't find a logical answer.

Answer 1

Since it is i.i.d., the future result will not depend on the past (see also a question on exchangeability here). Since the coin is fair, the probability to land head or tail is the same, $p=0.5$. As a consequence, if you predict head, the probability you get the prize is $0.5$. Same if you predict tail.

A whole answer on how you can(not) load a coin.

Update

While reading a recent entry in Larry Wasserman's blog I saw there a reference to a topic which would be relevant here, but I was totally unaware of. It is the online (sequential) learning and you can read more about again on Wasserman's blog.

Answer 2

In any type of classification or prediction, one should be aware of the costs different types of wrong answers and the benefits of different types of right answers. For instance, sometimes true positives are more valuable than true negatives, false positives might be more costly than false negatives, etc. This can lead to some interesting results and can be explored by considering the confusion matrix and an associated cost matrix and/or loss function.

However, things seem much simpler in this specific case. From the comments:

I just want to guess next one. If I can not guess correctly nothing happens, if I guess correct I will get prize.

This simplifies matters and no exotic cost matrix or anything of the sort is required. If the only thing that matters is accuracy (right answer or wrong answer), then one can simply guess the most likely answer. By definition of a fair coin, each side has a probability of $\frac{1}{2}$. So if you are forced to choose between the two, go with whatever tie-breaking strategy you feel comfortable with; either choice is as good as the other.

Note that this might change a bit if you do not start with the assumption that the coin is fair (are you sure this is valid?). Without that assumption, one might wonder whether one side is intrinsically more likely than the other. One might further wonder whether results are independent of past results, etc. However, in the scenario as described, it doesn't' really matter what you pick.

Answer 3

There is a series of papers in 'The Mathematical Scientist' that look at prediction of Bernoulli outcomes like this in cases where the mechanism is designed to be fair (i.e., IID with fixed probability = 1/2) but there is a possibility of bias. The basic idea is that if you believe the observable outcomes to be exchangeable, and you believe the bias to be exchangeable in direction, then you should predict whatever came up the most (the 'frequent outcome approach'). The papers are:

O'Neill (2015) Further results in binomial prediction. The Mathematical Scientist 40(1), pp. 13-22.

O'Neill (2012) Binomial prediction using the frequent outcome approach. The Mathematical Scientist 37(2), pp. 106-121.

O'Neill and Puza (2005) In defence of the reverse gambler's belief. The Mathematical Scientist 30(2), pp. 1-15.

Answer 4

If you know that it's a fair coin with i.i.d tosses then it doesn't matter which way you guess - that is one way of saying what "fair" means - and coins have no memory - that's one way of saying what i.i.d. means.

If, on the other hand, you are just told that it's a coin with i.i.d. tosses, then I'd guess heads because the evidence (slight as it is) is that heads are favored.

On yet the other hand, if you are told that it is fair but NOT i.i.d. (not sure how you could rig such a coin, but maybe there is a way) I'd guess tails because then the coin would somehow have a memory.

And if the coin was tossed by an expert at slight of hand magic, I wouldn't bet at all!

Answer 5

I have a literal answer, sorry......

You guys are way way smarter than I, but I am certain that all of you would also like to know the true way of consistently predicting a coin toss anway.

Having said that I rarely if ever lose a coin toss- No tricks (per say) Regardless of whom flips it.

First off, the coin isn't fair - it doesn't weigh the same on both sides and the weight isn't equally distributed across either face. The flipper is also biased- (I favor flipping while tails is up) and sometimes switch between catching over hand or underhand depending on what the other person calls. Not to mention the coin position pre-flip, finger to coin ratio preferences, the friction rate between the finger nail and the coin's surface, the horizontal orientation of the coin, thumb to forefinger resistantance etc... So flipping a coin is not static- force, speed, quantity and angle of rotations, height, atmospheric changes and a bunch of other variables that I am certain you can come up with will all affect the outcome; is the coin called pre-flip? Mid-Flip? Personally I like a lot of height, speed and making the "ting" noise. I prefer flipping right handed, catching the coin overhand, and then slapping the coin on the top of my other hand. The higher it goes the easier it is to predict- again no tricks...

A few other variables to consider; Is the coin caught underhand or overhand, does it land on the ground? Does it bounce off the ground? This affects the final outcome as well as you can get a "false" heads/tails. Statistics are also on the side of the person flipping who "wills" it to be heads or tails.

For my prediction control test, call it in the air, use the underhand catching method and don't "slap" the coin on the top of the other hand (this actually flips the coin again). Once you flip focus on the coin- there will be a predominating image in favor of heads or tails while in flight. If you see heads it will land heads up on the palm of your hand- if you consistantly see heads and get tails- call tails....etc

It takes a little practice but its super easy, I have gotten over thirty in a row right before being wrong and that's because I lose interest and stop paying attention. (or catch it poorly).

You can use the same principles when watching someone else flip- depending on their "style" you may have to call heads when you actually saw a tails image in flight to accurately get a heads.