A coin can only do one of two things. It can either come up heads or it can come up tails. As bets go, betting on the flip of a coin is pretty straightforward. Yet, in Stoppard's play, a coin toss becomes an immensely complicated thing. When you throw up that quarter, there are many more questions than whether or not it will come down heads or tails.
A coin toss is a classic example in classes on probability. It is one of the first things that comes to mind when we talk about 'chance': heads I win, tails you lose. In the first scene of the play, the coin comes down heads over one hundred times in a row. The chances of this happening are one in 2 to the 100th power. In other words, the chances are very, very small. All of a sudden, it seems that tossing a coin is no longer about chance, but about fate. This is what gets Guil so scared. It's like he is getting a sign from above, like he is seeing the Virgin Mary in the flesh. To him, it's that big.
Usually in books, getting "a sign from above" is a good thing, so why is Guil so scared? The reason is that this sign is ambiguous. It's unclear. One way to think about it: what if God sent you a message in Morse code and you didn't know Morse code? Guil tries to consider a bunch of different explanations for what is going on. One is, in fact, divine intervention. One of the more creative ideas he has is "Inside where nothing shows, I am the essence of a man spinning double-headed coins, and betting against himself in private atonement for the sins of an unremembered past" (6.56). From the very start of the play, the coin focuses in on major issues: are our lives controlled by chance or by fate? If they're controlled by fate, is there any way of knowing what that fate is? Either way, is there any hope of having free will?
This becomes very explicit in Act Two. After speaking with Hamlet, Ros and Guil try to figure out what is going on. Guil says something very important, so important that we're going to include the whole quote right here:
GUIL: Wheels have been set in motion, and they have their own pace, to which we are … condemned. Each move is dictated by the previous one – that is the meaning of order. If we start being arbitrary it'll just be a shambles: at least, let us hope so. Because if we happened, just happened to discover, or even suspect, that our spontaneity was part of their order, we'd know that we were lost. (He sits.) A Chinaman of the T'ang Dynasty – and, by which definition, a philosopher – dreamed he was a butterfly, and from that moment he was never quite sure that he was not a butterfly dreaming it was a Chinese philosopher. Envy him; in his two-fold security. (2.67)
There is no explicit mention of coins here, but there is a lot of talk about fate and order, and "two-fold security" seems to harken back to the way a coin can fall: either heads or tails. Just as someone who throws a coin up in the air knows that it will either fall heads or tails, the Chinaman knows that he is either one thing (a philosopher) or the other (a butterfly). He has only two options, and though he does not know which one is true, he can be reassured that at least he only has two options. Guil, by contrast, is afraid of things becoming too arbitrary. He is so uncertain of his situation that it almost feels like he is trapped in a whirlpool. If someone just told him, "Guil, you are either X or Y, but I can't tell you which one," then he would be immensely relieved.
Now what's all this business about "if we happened, just happened to discover, or even suspect, that our spontaneity was part of their order, we'd know that we were lost." Take some time and meditate on this one – this idea is far out. What Guil is saying is that we have to be able to believe that we can act spontaneously, that not everything in the world is ordered. In other words, we have to believe that we can act freely. If we don't have free will, life starts to seem really meaningless really fast. Go back through and read some of the banter between Ros and Guil: it's like they're trying to prove this line. They're trying to prove that they can act spontaneously, that they can act freely.
The coin is another way that this belief manifests itself. We believe that when we throw up a coin it is not already decided which way it will come down. There must be two possibilities! If we start to believe that the flip of a coin is already predetermined, then fate subsumes chance. What we mean is that all the things that we call "chance" are suddenly seen to be fate, just more subtly expressed. This is what gets Guil so freaked out in the beginning. It's starting to seem like there is no such thing as chance, which means that there is no such thing as free will.
There's one other thing about a coin: it's money. You bet on it because you are acting in your own self-interest. When Guil asks Ros what he would do if all of the coins had come down tails (so Guil would win instead of Ros), Ros says that he would check the coins. Guil then says, "I'm relieved. At least we can still count on self-interest as a predictable factor" (1.40). What this means in the context of our earlier discussion is that all of this stuff about fate and chance isn't just philosophical mumbo jumbo: it's very tied into what matters most to us – whether or not we are free. On another level, self-interest might play a role in the debate about fate and chance. Is it possible to act against one's own self-interest? If it's not, then is anyone really free?