Do miracles have a probability of zero, or are they simply extremely unlikely events?

Question

Splitting the sea into two seems physically impossible because they break physical laws. However, is this because the chance of atoms rearranging together such that the sea splits is extremely minute, or is this because it is physically impossible?

Another example might make this question more clear. Assume you have a fair coin. You flip the coin and it lands on heads 200,000 straight times. Is this an example of an extremely unlikely event or is this also an example of a physically impossible event?

How do we differentiate the two? I can potentially see arguments from both sides. Most books that I have read would class the latter event as a very unlikely event and the former event as a physically impossible event. But is it really physically possible for a fair coin to land on one side 200,000 straight times without some sort of influence causing it to be as such? It would beg the question as to how it landed on the same side for that long if it was fair.

Answer 1

Tossing a fair coin so that it lands on the same side 200,000 times is unlikely (that is an understatement!) but possible. After all, every time you toss the coin there is no law that says it cannot land on that side. An impossible event, on the other hand, is not a matter of chance. I could not swallow an ocean liner, no matter how many times I tried.

Answer 2

The answer to your question how we differentiate your two cases is that they are framed in different contexts. The first is framed in the context of our theories of physics and the second in the context of probability theory. In each case, the context is the reason for the specified classification.

You offer these cases as sharing a similarity, that they are both anomalous. This is true of the first, but not of the second.

In the case of the first, we have to ask whether it is more likely that the account of the event is false or that the event is contrary to the laws of physics. If we are forced to accept the account of the event, we label it anomalous and look for an explanation. We may find one that is compatible with our existing theories, but cannot rule out in advance the more remote possibility that a successful explanation may involve altering existing theories. (Hence, anomalous events can be very useful in scientific research.)

Given your mention of miracles in the heading of the question, you may like to consider that the probability of a given event being caused by divine intervention will always be less likely than it being the result of something else – even if we do not yet know what that something is. Miracles have been very popular evidence of God’s existence (and power). So we have to acknowledge that someone who believes in divine intervention irrespective of the laws of nature will estimate the probabilities differently. Clearly, there is here no argument, either for or against the existence of God. You might like to consult Miracles (Stanford Encyclopedia of Philosophy)

What you have missed in your specification of the second case is that the outcome 200,000 consecutive heads is no more anomalous that 100,000 heads every time followed by 100,000 tails every time, or 200,000 heads and tails alternating or any other sequence. To put it another way, if you bet on the outcome of a single toss 200,000 times, the odds of success are always 1:2. If you bet on the outcome of two tosses at a time, no matter what sequence you bet on, there are 4 outcomes, so the odds are 1:4. If you bet on the outcome of three throws, the odds are 1:8. And so on.

Answer 3

Do miracles have a probability of zero, or are they simply extremely unlikely events?

The smaller the sample set the greater the freedom to infer.

All estimates of the probability of "intelligent" life in the universe ignore a simple truth... We are the only "intelligent" life we are aware of. A sample set of one in the known universe (that we know of). That sounds like an extremely unlikely event and most would argue that intelligent life is a miracle. So a miracle is an event that is extremely unlikely and does not have a probability of zero.