I was going to start formalizing the concepts from Uncertainty Wednesday but I got a reader suggestion to do this in the context of a coin flipping example. This is what pretty much every introductory text uses because, well, coin flips are actually used in practice to decide things (especially in sports) and also coins have only two sides (heads and tails). It will be interesting to see if I can show how the Uncertainty Wednesday framework makes coin flipping more interesting than the way it is usually presented.

The setup: we observe a coin being flipped (by a machine or a human, we will get back to that) resulting in either Heads (H) or Tails (T).

In our framework, the reality is the coin flipping mechanism and the sequence of Hs and Ts are our observations about that reality. In addition, we have an explanation about how the coin flipping mechanism works.

Comment 1: Someone somewhere will surely ask, but what if the coin lands on its corner? We will simply make this go away by assuming the coin lands on a surface that jiggles a bit and so even if the coin were to land on its corner it would shortly thereafter fall to one side.

Comment 2: In the framework, this additional specification of removing the corner case is actually *reducing* the resolution of our observation. We are doing this here on purpose for simplicity but it is worth noting that’s what we are doing.

Comment 3: There are tons of other things that we could observe that we are *not* observing, such as how long a toss takes, what sound the toss makes, and so on. Not having observations other than the outcome of the toss could be for many reasons, such as we have chosen not to or we have been forbidden from doing so. Again this is a simplification, but forgetting it can easily lead on astray.

Comment 4: It is absolutely critical to understand that in this series we will *always* consider explanations and observations *together*. The fatal flaw in so much introduction to probability is that they entirely ignore the existence of explanations. Sequences of Heads and Tail are treated as outcomes, rather than as observations. My contention in Uncertainty Wednesday is that only by looking at the interaction between observations and explanations can we really understand what’s going on.

Comment 5: There are many possible explanations for any particular coin flipping reality. We will see that the explanation which has the highest uncertainty and contains the least information is that the mechanism is truly random and that in each coin flip we are equally likely to observe Heads or Tails (we will make all of this precise shortly). This contrasts with an explanation where, say a referee, has practiced flipping a coin to make Heads more likely, or has picked a fake coin that has Heads on both sides!

Comment 6: As we observe a sequence of Heads and Tails our uncertainty *may* be reduced both because we are having more observations but also because we revise our explanation! For instance, if we see a long sequence of only Heads, we may come to favor an explanation of a cheating referee. I say “may” because it could also be that we learn nothing new from the sequence.

So this is the coin flipping example. I presented in a condensed form relative to the four part Zoltar example, so if you are just starting on this series I would recommend you go back and read that also, as well as the initial framework post.

I was going to start formalizing the concepts from Uncertainty Wednesday but I got a reader suggestion to do this in the context of a coin flipping example. This is what pretty much every introductory text uses because, well, coin flips are actually used in practice to decide things (especially in sports) and also coins have only two sides (heads and tails). It will be interesting to see if I can show how the Uncertainty Wednesday framework makes coin flipping more interesting than the way it is usually presented.

The setup: we observe a coin being flipped (by a machine or a human, we will get back to that) resulting in either Heads (H) or Tails (T).

In our framework, the reality is the coin flipping mechanism and the sequence of Hs and Ts are our observations about that reality. In addition, we have an explanation about how the coin flipping mechanism works.

Comment 1: Someone somewhere will surely ask, but what if the coin lands on its corner? We will simply make this go away by assuming the coin lands on a surface that jiggles a bit and so even if the coin were to land on its corner it would shortly thereafter fall to one side.

Comment 2: In the framework, this additional specification of removing the corner case is actually *reducing* the resolution of our observation. We are doing this here on purpose for simplicity but it is worth noting that’s what we are doing.

Comment 3: There are tons of other things that we could observe that we are *not* observing, such as how long a toss takes, what sound the toss makes, and so on. Not having observations other than the outcome of the toss could be for many reasons, such as we have chosen not to or we have been forbidden from doing so. Again this is a simplification, but forgetting it can easily lead on astray.

Comment 4: It is absolutely critical to understand that in this series we will *always* consider explanations and observations *together*. The fatal flaw in so much introduction to probability is that they entirely ignore the existence of explanations. Sequences of Heads and Tail are treated as outcomes, rather than as observations. My contention in Uncertainty Wednesday is that only by looking at the interaction between observations and explanations can we really understand what’s going on.

Comment 5: There are many possible explanations for any particular coin flipping reality. We will see that the explanation which has the highest uncertainty and contains the least information is that the mechanism is truly random and that in each coin flip we are equally likely to observe Heads or Tails (we will make all of this precise shortly). This contrasts with an explanation where, say a referee, has practiced flipping a coin to make Heads more likely, or has picked a fake coin that has Heads on both sides!

Comment 6: As we observe a sequence of Heads and Tails our uncertainty *may* be reduced both because we are having more observations but also because we revise our explanation! For instance, if we see a long sequence of only Heads, we may come to favor an explanation of a cheating referee. I say “may” because it could also be that we learn nothing new from the sequence.

So this is the coin flipping example. I presented in a condensed form relative to the four part Zoltar example, so if you are just starting on this series I would recommend you go back and read that also, as well as the initial framework post.